SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Quasilinear boundary value problem with impulses: variational approach to resonance problem

Pavel Drábek12 and Martina Langerová2*

Author Affiliations

1 Department of Mathematics, University of West Bohemia, Univerzitní 22, Plzeň, 306 14, Czech Republic

2 NTIS, University of West Bohemia, Univerzitní 22, Plzeň, 306 14, Czech Republic

For all author emails, please log on.

Boundary Value Problems 2014, 2014:64  doi:10.1186/1687-2770-2014-64


We dedicate this paper to Professor Ivan Kiguradze for his merits in the theory of differential equations.


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/64


Received:5 December 2013
Accepted:7 March 2014
Published:24 March 2014

© 2014 Drábek and Langerová; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem.

MSC: 34A37, 34B37, 34F15, 49K35.

Keywords:
quasilinear impulsive differential equations; Landesman-Lazer condition; variational methods; critical point theory; linking theorem

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M1">View MathML</a> be a real number. We consider the homogeneous Dirichlet boundary value problem for one-dimensional p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M3">View MathML</a> is a spectral parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M5">View MathML</a>, is a given right-hand side.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M6">View MathML</a> be given points and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8">View MathML</a>, be given continuous functions. We are interested in the solutions of (1) satisfying the impulse conditions in the derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M9">View MathML</a>

(2)

For the sake of brevity, in further text we use the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M10">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a> this problem is considered in [1] where the necessary and sufficient condition for the existence of a solution of (1) and (2) is given. In fact, in the so-called resonance case, we introduce necessary and sufficient conditions of Landesman-Lazer type in terms of the impulse functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8">View MathML</a>, and the right-hand side f. They generalize the Fredholm alternative for linear problem (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a>.

In this paper we focus on a quasilinear equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15">View MathML</a> and look just for sufficient conditions. We point out that there are principal differences between the linear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a>) and the nonlinear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15">View MathML</a>). In the linear case, we could benefit from the Hilbert structure of an abstract formulation of the problem. It could be treated using the topological degree as a nonlinear compact perturbation of a linear operator. However, in the nonlinear case, completely different approach must be chosen in the resonance case. Our variational proof relies on the linking theorem (see [2]), but we have to work in a Banach space since the Hilbert structure is not suitable for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M15">View MathML</a>.

It is known that the eigenvalues of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M19">View MathML</a>

(3)

are simple and form an unbounded increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M20">View MathML</a> whose eigenspaces are spanned by functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M21">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22">View MathML</a> has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M23">View MathML</a> evenly spaced zeros in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M25">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M26">View MathML</a>. The reader is invited to see [[3], p.388], [[4], p.780] or [[5], pp.272-275] for further details. See also Example 1 below for more explicit form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M30">View MathML</a> , in (1). This is the nonresonance case. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4">View MathML</a>, there exists at least one solution of (1). In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a>, this solution is unique. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33">View MathML</a>, the uniqueness holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M34">View MathML</a>, but it may fail for certain right-hand sides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M36">View MathML</a>. See, e.g., [6] (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M37">View MathML</a>) and [7] (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M38">View MathML</a>).

The same argument as that used for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a> in [[1], Section 3] for the nonresonance case yields the following existence result for the quasilinear impulsive problem (1), (2).

Theorem 1 (Nonresonance case)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M30">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8">View MathML</a>, be continuous functions which are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44">View MathML</a>-subhomogeneous at ±∞, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M45">View MathML</a>

Then (1), (2) has a solution for arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4">View MathML</a>.

Variational approach to impulsive differential equations of the type (1), (2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a> was used, e.g., in paper [8]. The authors apply the mountain pass theorem to prove the existence of a solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M48">View MathML</a>. Our Theorem 1 thus generalizes [[8], Theorem 5.2] in two directions. Firstly, it allows also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M49">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M51">View MathML</a>) and, secondly, it deals with quasilinear equations (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33">View MathML</a>), too.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54">View MathML</a>. This is the resonance case. Contrary to the linear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M11">View MathML</a>), there is no Fredholm alternative for (1) in the nonlinear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33">View MathML</a>). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M57">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M58">View MathML</a>

is the sufficient condition for solvability of (1), but it is not necessary if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M33">View MathML</a>. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M60">View MathML</a> but f is ‘close enough’ to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M61">View MathML</a>, problem (1) has at least two distinct solutions. The reader is referred to [3] or [9] for more details. It appears that the situation is even more complicated for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M63">View MathML</a> (see, e.g., [10]).

In the presence of nonlinear impulses which have certain asymptotic properties (to be made precise below), we show that the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M64">View MathML</a> might still be the sufficient condition for the existence of a solution to (1) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a>) and (2). For this purpose we need some notation. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M66">View MathML</a> denote evenly spaced zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M69">View MathML</a> denote the union of intervals where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M70">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M71">View MathML</a>, respectively. We arrange <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M73">View MathML</a>, into three sequences: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M76">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M79">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M81">View MathML</a>. Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M83">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M84">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M85">View MathML</a>. The impulse condition (2) can be written in an equivalent form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M86">View MathML</a>

(4)

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M76">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M89">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M81">View MathML</a>, are continuous, bounded functions and there exist limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M92">View MathML</a>. We consider the following Landesman-Lazer type conditions: either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M93">View MathML</a>

(5)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M94">View MathML</a>

(6)

Our main result is the following.

Theorem 2 (Resonance case)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96">View MathML</a>in (1). Let the nonlinear bounded impulse functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M8">View MathML</a>, and the right-hand side<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M4">View MathML</a>satisfy either (5) or (6). Then (1), (2) has a solution.

The result from Theorem 2 is illustrated in the following special example.

Example 1 It follows from the first integral associated with the equation in (3) that the eigenvalues and the eigenfunctions of (3) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M107">View MathML</a>, see [[3], p.388]. Let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M108">View MathML</a> in (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M113">View MathML</a>, in (2). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M115">View MathML</a>, condition (6) reads as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M116">View MathML</a>

2 Functional framework

We say that u is the classical solution of (1), (2) if the following conditions are fulfilled:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M119">View MathML</a> is absolutely continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121">View MathML</a>;

• the equation in (1) holds a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M123">View MathML</a>;

• one-sided limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M125">View MathML</a> exist finite and (2) holds.

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M126">View MathML</a> is a weak solution of (1), (2) if the integral identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M127">View MathML</a>

(7)

holds for any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M128">View MathML</a>.

Integration by parts and the fundamental lemma in calculus of variations (see [[11], Lemma 7.1.9]) yields that every weak solution of (1), (2) is also a classical solution and vice versa. Indeed, let u be a weak solution of (1), (2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M129">View MathML</a> (the space of smooth functions with a compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M132">View MathML</a> elsewhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M24">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M134">View MathML</a>

Since v is arbitrary, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M135">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M136">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M137">View MathML</a> is absolutely continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M120">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M139">View MathML</a>

(8)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121">View MathML</a>. Taking now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M128">View MathML</a> arbitrary, integrating by parts in the first integral in (7) and using (8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M143">View MathML</a>

and hence also (2) follows. Similarly, we show that every classical solution is a weak solution at the same time.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M144">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a> be the dual of X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M147">View MathML</a> be the duality pairing between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a> and X. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M150">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M151">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M152">View MathML</a>

Lemma 1The operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M153">View MathML</a>have the following properties:

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M154">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44">View MathML</a>-homogeneous, odd, continuously invertible, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M156">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149">View MathML</a>.

(B) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M158">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44">View MathML</a>-homogeneous, odd and compact.

(J) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M160">View MathML</a>is bounded and compact.

By the linearity of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M162">View MathML</a>is a fixed element.

Proof See [[12], Lemma 10.3, p.120]. □

With this notation in hands we can look for (classical) solutions of (1), (2) either as for solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M149">View MathML</a> of the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M164">View MathML</a>

(9)

or, alternatively, as for critical points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M165">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M166">View MathML</a>

(10)

As mentioned already above, in the nonresonance case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96">View MathML</a>), we can use the Leray-Schauder degree argument and prove the existence of a solution of the equation (9) exactly as in [[1], proof of Thm. 1]. Note that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M44">View MathML</a>-subhomogeneous condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a> is used here instead of the sublinear condition imposed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a> in [1] and the proof of Theorem 1 follows the same lines. For this reason we skip it and concentrate on the resonance case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54">View MathML</a>) in the next section.

3 Resonance problem, variational approach

We use the following definition of linked sets and the linking theorem (cf.[13]).

Definition 1 Let ℰ be a closed subset of X and let Q be a submanifold of X with relative boundary ∂Q. We say that ℰ and ∂Q link if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M174">View MathML</a> and

(ii) for any continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M175">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M176">View MathML</a>, there holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M177">View MathML</a>.

(See [[14], Def. 8.1, p.116].)

Theorem 3 (Linking theorem)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M178">View MathML</a>satisfies the Palais-Smale condition. Consider a closed subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M179">View MathML</a>and a submanifold<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M180">View MathML</a>with relative boundary∂Q, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M181">View MathML</a>. Suppose thatand∂Qlink in the sense of Definition 1, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M182">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M183">View MathML</a>is a critical value of ℱ.

(See [[14], Thm. 8.4, p.118].)

The purpose of the following series of lemmas is to show that the hypotheses of Theorem 3 are satisfied provided that either (5) or (6) holds. From now on we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M53">View MathML</a> (for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M54">View MathML</a>) in (1).

Lemma 2If either (5) or (6) is satisfied, thensatisfies the Palais-Smale condition.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M186">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M188">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a>. We must show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M190">View MathML</a> has a subsequence that converges in X. We prove first that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M191">View MathML</a> is a bounded sequence. We proceed via contradiction and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M192">View MathML</a> and consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M193">View MathML</a>. Without loss of generality, we can assume that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M194">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M195">View MathML</a> (weakly) in X (X is a reflexive Banach space). Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M196">View MathML</a>

dividing through by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M197">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M198">View MathML</a>

By the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M160">View MathML</a> we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M200">View MathML</a>. We also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M201">View MathML</a>. By the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M158">View MathML</a> we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M203">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M205">View MathML</a> in X by Lemma 1(A). It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M206">View MathML</a>.

We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M207">View MathML</a> and remark that a similar argument follows if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M208">View MathML</a>. Next we estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M209">View MathML</a>

(11)

Our assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M187">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M211">View MathML</a>

(12)

and the Cauchy-Schwarz inequality implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M212">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M213">View MathML</a> denotes the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a>. It follows from (11)-(13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M215">View MathML</a>

Dividing through by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M216">View MathML</a> and writing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M217">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M218">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M220">View MathML</a>

(14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M221">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222">View MathML</a>, we obtain from (14):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M223">View MathML</a>

(15)

Recall that X embeds compactly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M224">View MathML</a>, so, without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M121">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M227">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M228">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M229">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M230">View MathML</a> as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M231">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222">View MathML</a> by an application of the l’Hospital rule to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M233">View MathML</a>. Notice that by the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M235">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72">View MathML</a> is a zero point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M238">View MathML</a>. Thus, passing to the limit in (15) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M240">View MathML</a>

which contradicts (5) or (6). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M241">View MathML</a> is bounded.

By compactness there is a subsequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M243">View MathML</a> converge in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a> (see Lemma 1(B), (J)). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M188">View MathML</a> by our assumption, we also have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M246">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M146">View MathML</a>. Finally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M248">View MathML</a> converges in X by Lemma 1(A). The proof is finished. □

With the Palais-Smale condition in hands, we can turn our attention to the geometry of the functional ℱ. To this end we have to find suitable sets which link in the sense of Definition 1. Actually, we use the sets constructed in [13] and explain that they fit with the hypotheses of Theorem 3 if either (5) or (6) is satisfied.

Consider the even functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M249">View MathML</a>

and the manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M250">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M96">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M252">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M253">View MathML</a> represents the unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M254">View MathML</a>. Next we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M255">View MathML</a>

(16)

It is proved in [[15], Section 3] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M256">View MathML</a> is a sequence of eigenvalues of homogeneous problem (3). It then follows from the results in [16] that this sequence exhausts the set of all eigenvalues of (3) with the properties described in Section 1.

Now consider the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M257">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M258">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M259">View MathML</a> is a characteristic function of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M260">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M261">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M262">View MathML</a> is symmetric and is homeomorphic to the unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M263">View MathML</a>. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M264">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M265">View MathML</a>

Notice that the second equality holds thanks to the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M266">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M268">View MathML</a>, while the third one follows from the p-homogeneity of B. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M269">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M270">View MathML</a>. A similar computation then shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M271">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M264">View MathML</a>. For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M273">View MathML</a>, we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M274">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M275">View MathML</a> is homeomorphic to the closed unit ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M254">View MathML</a>. For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M277">View MathML</a>, we denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M278">View MathML</a>

a super-level set, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M279">View MathML</a>

The existence of a pseudo-gradient vector field with the following properties is proved in [[13], Lemma 6] (cf. [[14], pp.77-79] and [[2], p.55]).

Lemma 3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M280">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M281">View MathML</a>and a one-parameter family of homeomorphisms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M282">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M283">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M284">View MathML</a>or if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M285">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M286">View MathML</a>is strictly decreasing intif<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M287">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M288">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M289">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M290">View MathML</a>.

An important fact is that the flow η ‘lowers’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M275">View MathML</a> and ‘raises’ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M292">View MathML</a> if we modify them as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M293">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M294">View MathML</a>

Then, by Lemma 3 and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M295">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M296">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M297">View MathML</a> with equality if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M298">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M299">View MathML</a>. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M300">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M301">View MathML</a> with equality if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M302">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M299">View MathML</a>.

It is proved in [[13], Lemma 7] that the couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M305">View MathML</a> satisfies condition (ii) from Definition 1. It is also proved in [[13], Lemma 8] that the couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M306">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M307">View MathML</a> satisfies the same condition. To show that also other hypotheses of Theorem 3 are satisfied, we need some technical lemmas.

Lemma 4If (6) is satisfied, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M308">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M310">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M311">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312">View MathML</a>.

Proof We proceed via contradiction and assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M314">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M315">View MathML</a>

(17)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316">View MathML</a> is compact, we may assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M317">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M319">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M320">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M321">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M322">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M323">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M324">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M325">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M326">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M324">View MathML</a>. However, this contradicts (17).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M328">View MathML</a>, we still have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M329">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M330">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M331">View MathML</a>. The boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M333">View MathML</a>, and uniform convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M334">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M335">View MathML</a> (due to continuous embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M336">View MathML</a>) then yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M337">View MathML</a>

by the first inequality in (6). This contradicts (17) again. Notice that by the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M12">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M339">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M72">View MathML</a> is a zero point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M22">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M238">View MathML</a>. The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M343">View MathML</a> is proved similarly using the second inequality in (6). □

Lemma 5If (6) is satisfied, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M344">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M345">View MathML</a>

(18)

Proof There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M346">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M347">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M348">View MathML</a>

By Lemma 4 there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M277">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M350">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M352">View MathML</a>

Thus there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M353">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M354">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M355">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M312">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M357">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M358">View MathML</a> and (18) is proved. □

Now we can finish the proof of Theorem 2 under assumption (6). Indeed, it follows from (18) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M359">View MathML</a> and thus the hypotheses of Theorem 3 hold with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M360">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M305">View MathML</a>. It then follows that ℱ has a critical point and hence (1), (2) has a solution.

Next we show that the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M362">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M363">View MathML</a> satisfy the hypotheses of Theorem 3 if (5) is satisfied.

The principal difference consists in the fact that, in contrast with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M316">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M365">View MathML</a> is not compact. That is why one more technical lemma is needed.

Lemma 6For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M366">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M368">View MathML</a>

(19)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M369">View MathML</a>. (Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M370">View MathML</a>is the ball inXcentered at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M371">View MathML</a>with radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M372">View MathML</a>.)

Proof We note that the pseudo-gradient flow η from Lemma 3 is constructed as a solution of the initial value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M373">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M374">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M375">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M376">View MathML</a> is a locally Lipschitz continuous symmetric pseudo-gradient vector field associated with E on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M377">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M378">View MathML</a> is a smooth function such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M379">View MathML</a> for u satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M380">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M381">View MathML</a> for u satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M382">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M383">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M384">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M385">View MathML</a>. Without loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M386">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M387">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M388">View MathML</a>. Observe that there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M389">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M390">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M391">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M392">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M393">View MathML</a>. Since E satisfies the Palais-Smale condition on (see [[13], Lemma 2]), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M395">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M396">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M397">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M398">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M393">View MathML</a>. The last but one inequality holds due to the following property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M376">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M401">View MathML</a>

(see [14] and [2]). We also used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M402">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M403">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M404">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M405">View MathML</a>. □

The following lemma is a counterpart of Lemma 4 in the case of condition (5).

Lemma 7If (5) is satisfied, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M308">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M309">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M408">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M311">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M410">View MathML</a>.

Proof We proceed via contradiction and assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M412">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M413">View MathML</a>

(20)

If there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M414">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M415">View MathML</a> for all k large enough, then Lemma 6 leads to the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M416">View MathML</a>

contradicting (20). Thus it must be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M417">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M222">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M334">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M335">View MathML</a>, we still have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M421">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M422">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M331">View MathML</a>. Similar arguments as in the proof of Lemma 4 lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M424">View MathML</a>

by the second inequality in (5). This contradicts (20) again. The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M425">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M426">View MathML</a> is proved similarly but using the first inequality in (5). □

Lemma 8If (5) is satisfied, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M344">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M428">View MathML</a>

(21)

Proof By Lemma 7 there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M429">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M430">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M431">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M432">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M346">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M434">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M435">View MathML</a>

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M436">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M437">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M438">View MathML</a>

Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M273">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M440">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M441">View MathML</a>

and (21) is proved. □

It follows that the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M362">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M363">View MathML</a> satisfy the hypotheses of Theorem 3 if (5) is satisfied. The proof of Theorem 2 is thus completed.

Final remark Reviewers of our manuscript suggested to include some recent references on impulsive problems. Variational approach to impulsive problems can be found, e.g., in [17-21]. The last reference deals with the p-Laplacian with the variable exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M444">View MathML</a>. Singular impulsive problems are treated in [22-24]. Impulsive problems are still ‘hot topic’ attracting the attention of many mathematicians and the bibliography on that topic is vast.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors read and approved the final manuscript.

Acknowledgements

This research was supported by Grant 13-00863S of the Grant Agency of Czech Republic and by the European Regional Development Fund (ERDF), project ‘NTIS - New Technologies for the Information Society’, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

References

  1. Drábek, P, Langerová, M: On the second order equations with nonlinear impulses - Fredholm alternative type results. Topol. Methods Nonlinear Anal. (to appear)

  2. Ghoussoub, N: Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, Cambridge (1993)

  3. del Pino, M, Drábek, P, Manásevich, R: The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian. J. Differ. Equ.. 151, 386–419 (1999). Publisher Full Text OpenURL

  4. Drábek, P, Manásevich, R: On the closed solutions to some nonhomogeneous eigenvalue problems with p-Laplacian. Differ. Integral Equ.. 12, 773–788 (1999)

  5. Lindqvist, P: Some remarkable sine and cosine functions. Ric. Mat.. XLIV, 269–290 (1995)

  6. del Pino, M, Elgueta, M, Manásevich, R: A homotopic deformation along p of a Leray-Schauder degree result and existence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M447">View MathML</a>. J. Differ. Equ.. 80, 1–13 (1989). Publisher Full Text OpenURL

  7. Fleckinger, J, Hernández, J, Takáč, P, de Thélin, F: Uniqueness and positivity for solutions of equations with the p-Laplacian. In: Caristi G, Mitidieri E (eds.) (1995)

  8. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text OpenURL

  9. Drábek, P, Girg, P, Takáč, P, Ulm, M: The Fredholm alternative for the p-Laplacian: bifurcation from infinity, existence and multiplicity. Indiana Univ. Math. J.. 53, 433–482 (2004). Publisher Full Text OpenURL

  10. Manásevich, R, Takáč, P: On the Fredholm alternative for the p-Laplacian in one dimension. Proc. Lond. Math. Soc.. 84, 324–342 (2002). Publisher Full Text OpenURL

  11. Drábek, P, Milota, J: Methods of Nonlinear Analysis, Applications to Differential Equations, Springer, Basel (2013)

  12. Drábek, P: Solvability and Bifurcations of Nonlinear Equations, Longman, Harlow (1992)

  13. Drábek, P, Robinson, SB: Resonance problems for the one-dimensional p-Laplacian. Proc. Am. Math. Soc.. 128, 755–765 (1999)

  14. Struwe, M: Variational Methods; Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, New York (1990).

  15. Drábek, P, Robinson, SB: Resonance problems for the p-Laplacian. J. Funct. Anal.. 169, 189–200 (1999). Publisher Full Text OpenURL

  16. Drábek, P, Robinson, SB: On the generalization of the Courant nodal domain theorem. J. Differ. Equ.. 181, 58–71 (2001)

  17. Chen, H, He, Z: Variational approach to some damped Dirichlet problems with impulses. Math. Methods Appl. Sci.. 36(18), 2564–2575 (2013). Publisher Full Text OpenURL

  18. Otero-Espinar, V, Pernas-Castaño, T: Variational approach to second-order impulsive dynamic equations on time scales. Bound. Value Probl.. 2013, Article ID 119 (2013)

  19. Xiao, J, Nieto, JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst.. 348(2), 369–377 (2011). Publisher Full Text OpenURL

  20. Galewski, M: On variational impulsive boundary value problems. Cent. Eur. J. Math.. 10(6), 1969–1980 (2012). Publisher Full Text OpenURL

  21. Galewski, M, O’Regan, D: Impulsive boundary value problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/64/mathml/M451">View MathML</a>-Laplacian’s via critical point theory. Czechoslov. Math. J.. 62(4), 951–967 (2012). Publisher Full Text OpenURL

  22. Sun, J, Chu, J, Chen, H: Periodic solution generated by impulses for singular differential equations. J. Math. Anal. Appl.. 404(2), 562–569 (2013). Publisher Full Text OpenURL

  23. Chu, J, Nieto, JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc.. 40(1), 143–150 (2008). Publisher Full Text OpenURL

  24. Sun, J, O’Regan, D: Impulsive periodic solutions for singular problems via variational methods. Bull. Aust. Math. Soc.. 86(2), 193–204 (2012). Publisher Full Text OpenURL