This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Multiplicity of positive solutions for eigenvalue problems of ( p , 2 ) -equations

Leszek Gasiński1* and Nikolaos S Papageorgiou2

Author Affiliations

1 Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, ul. Łojasiewicza 6, Kraków, 30-348, Poland

2 Department of Mathematics, National Technical University, Zografou Campus, Athens, 15780, Greece

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Boundary Value Problems 2012, 2012:152  doi:10.1186/1687-2770-2012-152


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/152


Received:13 September 2012
Accepted:7 December 2012
Published:28 December 2012

© 2012 Gasiński and Papageorgiou; 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 cited.

Abstract

We consider a nonlinear parametric equation driven by the sum of a p-Laplacian (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2">View MathML</a>) and a Laplacian (a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1">View MathML</a>-equation) with a Carathéodory reaction, which is strictly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4">View MathML</a>-sublinear near +∞. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem for the nonlinear eigenvalue problem. So, we show that there is a critical parameter value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6">View MathML</a> the problem has at least two positive solutions, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7">View MathML</a>, then the problem has at least one positive solution and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8">View MathML</a>, it has no positive solutions.

MSC: 35J25, 35J92.

Keywords:
nonlinear regularity; tangency principle; p-Laplacian; bifurcation-type theorem; positive solutions

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M9">View MathML</a> be a bounded domain with a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M10">View MathML</a>-boundary Ω. In this paper, we study the following nonlinear Dirichlet eigenvalue problem:

Here, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M12">View MathML</a> we denote the p-Laplace differential operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M13">View MathML</a>

(with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M14">View MathML</a>). In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a> is a parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M17">View MathML</a> is a Carathéodory function (i.e., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M19">View MathML</a> is measurable and for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M21">View MathML</a> is continuous), which exhibits strictly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4">View MathML</a>-sublinear growth in the ζ-variable near +∞. The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a>. So, we prove a bifurcation-type theorem, which establishes the existence of a critical parameter value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6">View MathML</a>, problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> has at least two nontrivial positive smooth solutions, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7">View MathML</a>, problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> has at least one nontrivial positive smooth solution and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8">View MathML</a>, problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> has no positive solution. Similar nonlinear eigenvalue problems with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4">View MathML</a>-sublinear reaction were studied by Maya and Shivaji [1] and Rabinowitz [2] for problems driven by the Laplacian and by Guo [3], Hu and Papageorgiou [4] and Perera [5] for problems driven by the p-Laplacian. However, none of the aforementioned works produces the precise dependence of the set of positive solutions on the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a> (i.e., they do not prove a bifurcation-type theorem). We mention that in problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> the differential operator is not homogeneous in contrast to the case of the Laplacian and p-Laplacian. This fact is the source of difficulties in the study of problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> which lead to new tools and methods.

We point out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1">View MathML</a>-equations (i.e., equations in which the differential operator is the sum of a p-Laplacian and a Laplacian) are important in quantum physics in the search for solitions. We refer to the work of Benci, D’Avenia-Fortunato and Pisani [6]. More recently, there have been some existence and multiplicity results for such problems; see Cingolani and Degiovanni [7], Sun [8]. Finally, we should mention the recent papers of Marano and Papageorgiou [9,10]. In [9] the authors deal with parametric p-Laplacian equations in which the reaction exhibits competing nonlinearities (concave-convex nonlinearity). In [10], they study a nonparametric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M36">View MathML</a>-equation with a reaction that has different behavior both at ±∞ and at 0 from those considered in the present paper, and so the geometry of the problem is different.

Out approach is variational based on the critical point theory, combined with suitable truncation and comparison techniques. In the next section, for the convenience of the reader, we briefly recall the main mathematical tools that we use in this paper.

2 Mathematical background

Let X be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M37">View MathML</a> be its topological dual. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M38">View MathML</a> we denote the duality brackets for the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M39">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40">View MathML</a>. A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M41">View MathML</a> is a critical point of φ if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M42">View MathML</a>. A number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M43">View MathML</a> is a critical value of φ if there exists a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M41">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M45">View MathML</a>.

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40">View MathML</a> satisfies the Palais-Smale condition if the following is true:

‘Every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M47">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M48">View MathML</a> is bounded and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M49">View MathML</a>

admits a strongly convergent subsequence.’

This compactness-type condition is crucial in proving a deformation theorem which in turn leads to the minimax theory of certain critical values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40">View MathML</a> (see, e.g., Gasinski and Papageorgiou [11]). A well-written discussion of this compactness condition and its role in critical point theory can be found in Mawhin and Willem [12]. One of the minimax theorems needed in the sequel is the well-known ‘mountain pass theorem’.

Theorem 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M40">View MathML</a>satisfies the Palais-Smale condition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M53">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M54">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M55">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M56">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M57">View MathML</a>andcis a critical value ofφ.

In the analysis of problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>, in addition to the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59">View MathML</a>, we will also use the Banach space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M60">View MathML</a>

This is an ordered Banach space with a positive cone:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M61">View MathML</a>

This cone has a nonempty interior given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M62">View MathML</a>

where by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M63">View MathML</a> we denote the outward unit normal on Ω.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M64">View MathML</a> be a Carathéodory function with subcritical growth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M66">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M69">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M70">View MathML</a>

(the critical Sobolev exponent).

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M71">View MathML</a>

and consider the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M73">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M74">View MathML</a>

(2.1)

The next proposition is a special case of a more general result proved by Gasinski and Papageorgiou [13]. We mention that the first result of this type was proved by Brezis and Nirenberg [14].

Proposition 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75">View MathML</a>is defined by (2.1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76">View MathML</a>is a local<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M77">View MathML</a>-minimizer of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75">View MathML</a>, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M79">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M80">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M81">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83">View MathML</a>is also a local<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59">View MathML</a>-minimizer of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M75">View MathML</a>, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M86">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M87">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M88">View MathML</a>. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89">View MathML</a> if for all compact subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M90">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M91">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M92">View MathML</a>

Clearly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M94">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89">View MathML</a>. A slight modification of the proof of Proposition 2.6 of Arcoya and Ruiz [15] in order to accommodate the presence of the extra linear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M97">View MathML</a> leads to the following strong comparison principle.

Proposition 2.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M102">View MathML</a>are solutions of the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M103">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M104">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M105">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M106">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M107">View MathML</a>) be a nonlinear map defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M108">View MathML</a>

(2.2)

The next proposition can be found in Dinca, Jebelean and Mawhin [16] and Gasiński and Papageorgiou [11].

Proposition 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M109">View MathML</a> (where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M110">View MathML</a>) is defined by (2.2), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M111">View MathML</a>is continuous, strictly monotone (hence maximal monotone too), bounded and of type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M112">View MathML</a>, i.e., if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M113">View MathML</a>weakly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M114">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M115">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M113">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M118">View MathML</a>, then we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M119">View MathML</a>.

In what follows, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M120">View MathML</a> we denote the first eigenvalue of the negative Dirichlet p-Laplacian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M121">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M122">View MathML</a> and it admits the following variational characterization:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M123">View MathML</a>

(2.3)

Finally, throughout this work, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M124">View MathML</a> we denote the norm of the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59">View MathML</a>. By virtue of the Poincaré inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M126">View MathML</a>

The notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M124">View MathML</a> will also be used to denote the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M128">View MathML</a>. No confusion is possible since it will always be clear from the context which norm is used. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M18">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M130">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M132">View MathML</a>. We know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M133">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M134">View MathML</a> is superpositionally measurable (for example, a Carathéodory function), then we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M135">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M136">View MathML</a> we denote the Lebesgue measure on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M128">View MathML</a>.

3 Positive solutions

The hypotheses on the reaction f are the following.

H: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M138">View MathML</a> is a Carathéodory function such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M139">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M141">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M143">View MathML</a> and

(i) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M144">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M145">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M146">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M147">View MathML</a> uniformly for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M149">View MathML</a> uniformly for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>;

(iv) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M144">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152">View MathML</a> such that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M154">View MathML</a> is nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M155">View MathML</a>;

(v) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M156">View MathML</a>

then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M158">View MathML</a>

Remark 3.1 Since we are looking for positive solutions and hypotheses H concern only the positive semiaxis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M159">View MathML</a>, we may and will assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M160">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M162">View MathML</a>. Hypothesis H(ii) implies that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M164">View MathML</a> is strictly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4">View MathML</a>-sublinear near +∞. Hypothesis H(iv) is much weaker than assuming the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M164">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>.

Example 3.2 The following functions satisfy hypotheses H (for the sake of simplicity, we drop the z-dependence):

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M169">View MathML</a>. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M170">View MathML</a> is not monotone.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M171">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M172">View MathML</a> be the set of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>. We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M174">View MathML</a>

(if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M175">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M176">View MathML</a>).

Proposition 3.3If hypotheses H hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M177">View MathML</a>

Proof Clearly, the result is true if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M175">View MathML</a>. So, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180">View MathML</a>. So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M181">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M182">View MathML</a>

From Ladyzhenskaya and Uraltseva [[17], p.286], we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M183">View MathML</a>. Then we can apply Theorem 1 of Lieberman [18] and have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M184">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M185">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152">View MathML</a> be as postulated by hypothesis H(iv). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M187">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M188">View MathML</a>

From the strong maximum principle of Pucci and Serrin [[19], p.34], we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M189">View MathML</a>

So, we can apply the boundary point theorem of Pucci and Serrin [[19], p.120] and have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M190">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M191">View MathML</a>.

By virtue of hypotheses H(ii) and (iii), we see that we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M193">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M195">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M196">View MathML</a>. Then from the first part of the proof, we know that we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M197">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M198">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M199">View MathML</a>

(see (3.1) and recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M200">View MathML</a>), which contradicts (2.3). Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M201">View MathML</a>. □

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M203">View MathML</a> be the energy functional for problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M205">View MathML</a>

Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M206">View MathML</a>.

Proposition 3.4If hypotheses H hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179">View MathML</a>.

Proof By virtue of hypotheses H(i) and (ii), for a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M209">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M210">View MathML</a>

(3.2)

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M213">View MathML</a>

(3.3)

(see (3.2) and (2.3)).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M214">View MathML</a>. Then from (3.3) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215">View MathML</a> is coercive. Also, exploiting the compactness of the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M216">View MathML</a> (by the Sobolev embedding theorem), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215">View MathML</a> is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M219">View MathML</a>

(3.4)

Consider the integral functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M220">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M221">View MathML</a>

Hypothesis H(v) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M222">View MathML</a> and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M223">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a>, all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M162">View MathML</a>, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M226">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M59">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M226">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M231">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M232">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M16">View MathML</a> large, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M234">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M235">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M236">View MathML</a>

(see (3.4)), hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M237">View MathML</a>. From (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M238">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M239">View MathML</a>

(3.5)

On (3.5), we act with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M240">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M241">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M237">View MathML</a>.

From (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M244">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M245">View MathML</a> (see Proposition 3.3).

So, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M246">View MathML</a> big, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M247">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M179">View MathML</a>. □

Proposition 3.5If hypotheses H hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M250">View MathML</a>.

Proof Since by hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M180">View MathML</a>, we can find a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M252">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a> (see Proposition 3.3). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M254">View MathML</a> and consider the following truncation of the reaction in problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M255">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M256">View MathML</a>

(3.6)

This is a Carathéodory function. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M257">View MathML</a>

and consider the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M259">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M260">View MathML</a>

As in the proof of Proposition 3.4, using hypotheses H(i) and (ii), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M261">View MathML</a> is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M262">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M263">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M264">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M265">View MathML</a>

(3.7)

On (3.7) we act with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M266">View MathML</a>. Then

(see (3.6) and use the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269">View MathML</a>), so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M270">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M271">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M272">View MathML</a>.

Therefore, (3.7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M273">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M274">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M275">View MathML</a>. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M250">View MathML</a>. □

Proposition 3.6If hypotheses H hold, then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6">View MathML</a>problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>has at least two positive solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M279">View MathML</a>

Proof Note that Proposition 3.5 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M280">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M281">View MathML</a>. Then we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M283">View MathML</a>. We have

(3.8)

(3.9)

(recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M287">View MathML</a>). As in the proof of Proposition 3.5, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M288">View MathML</a>. We introduce the following truncation of the reaction in problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M290">View MathML</a>

(3.10)

This is a Carathéodory function. We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M291">View MathML</a>

and consider the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M72">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M293">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M294">View MathML</a>

It is clear from (3.10) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M295">View MathML</a> is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M297">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M298">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M299">View MathML</a>

(3.11)

Acting on (3.11) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M300">View MathML</a> and next with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M301">View MathML</a> (similarly as in the proof of Proposition 3.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M302">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M303">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M304">View MathML</a>.

Then (3.11) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M305">View MathML</a>

(see (3.10)), so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M306">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M307">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M308">View MathML</a> (recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M310">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M311">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M312">View MathML</a>

So, we can apply the tangency principle of Pucci and Serrin [[19], p.35] and infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M313">View MathML</a>

(3.12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M314">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M152">View MathML</a> be as postulated by hypothesis H(iv). Then

(see hypothesis H(iv) and use the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M317">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M269">View MathML</a>), so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M319">View MathML</a>

(3.13)

(see (3.12) and Proposition 2.3).

In a similar fashion, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M320">View MathML</a>

(3.14)

From (3.13) and (3.14), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M321">View MathML</a>

(3.15)

From (3.10), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M322">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M323">View MathML</a>.

So, (3.15) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83">View MathML</a> is a local <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M325">View MathML</a>-minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215">View MathML</a>. Invoking Proposition 2.3, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M327">View MathML</a>

(3.16)

Hypotheses H(i), (ii) and (iii) imply that for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M329">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M330">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M331">View MathML</a>

(3.17)

Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M131">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M333">View MathML</a>

(3.18)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M334">View MathML</a> (see (3.17) and (2.3)).

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M335">View MathML</a>. Then, from (3.18) and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M329">View MathML</a>, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M83">View MathML</a> is a local minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215">View MathML</a>. Without any loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M339">View MathML</a> (the analysis is similar if the opposite inequality holds). By virtue of (3.16), as in Gasinski and Papageorgiou [20] (see the proof of Theorem 2.12), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M340">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M341">View MathML</a>

(3.19)

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M215">View MathML</a> is coercive, hence it satisfies the Palais-Smale condition. This fact and (3.19) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M343">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M344">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M345">View MathML</a>

(3.21)

From (3.20) and (3.19), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M346">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M347">View MathML</a>. From (3.21), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M348">View MathML</a>. □

Next, we examine what happens at the critical parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M349">View MathML</a>.

Proposition 3.7If hypotheses H hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M350">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M351">View MathML</a> be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M352">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M353">View MathML</a>

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M354">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M355">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M356">View MathML</a>

(3.22)

We claim that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357">View MathML</a> is bounded. Arguing indirectly, suppose that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357">View MathML</a> is unbounded. By passing to a suitable subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M359">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M360">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M362">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M354">View MathML</a>. From (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M364">View MathML</a>

(3.23)

Recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M365">View MathML</a>

(see (3.1)), so the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M366">View MathML</a> is bounded. This fact and hypothesis H(ii) imply that at least for a subsequence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M367">View MathML</a>

(3.24)

(see Gasinski and Papageorgiou [20]). Also, passing to a subsequence if necessary, we may assume that

(3.25)

(3.26)

On (3.23) we act with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M370">View MathML</a>, pass to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371">View MathML</a> and use (3.24) and (3.26). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M372">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M373">View MathML</a>

Using Proposition 2.4, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M374">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M375">View MathML</a>

(3.27)

Passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371">View MathML</a> in (3.23) and using (3.24), (3.27) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M378">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M379">View MathML</a>, which contradicts (3.27).

This proves that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M357">View MathML</a> is bounded. So, passing to a subsequence if necessary, we may assume that

(3.28)

(3.29)

On (3.22) we act with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M383">View MathML</a>, pass to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M371">View MathML</a> and use (3.28) and (3.29). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M385">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M386">View MathML</a>

(since A is monotone) and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M387">View MathML</a>

(3.30)

(see Proposition 2.4).

Therefore, if in (3.22) we pass to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M388">View MathML</a> and use (3.30), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M389">View MathML</a>

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M390">View MathML</a> is a solution of problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M391">View MathML</a>.

We need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M392">View MathML</a>. From (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M393">View MathML</a>

From Ladyzhenskaya and Uraltseva [[17], p.286], we know that we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M394">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M395">View MathML</a>

Then applying Theorem 1 of Lieberman [18], we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M397">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M398">View MathML</a>

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M399">View MathML</a> is embedded compactly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M325">View MathML</a>. So, by virtue of (3.28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M401">View MathML</a>

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M402">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M403">View MathML</a>

(3.31)

Hypothesis H(iii) implies that for a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M208">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M405">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M406">View MathML</a>

(3.32)

From (3.31), it follows that we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M407">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M408">View MathML</a>

(3.33)

Therefore, for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M20">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M410">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M411">View MathML</a>

(see (3.32) and (3.33)), so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M412">View MathML</a>

(see (2.3)), thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M413">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M414">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M415">View MathML</a> to get a contradiction. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M392">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M417">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M350">View MathML</a>. □

The bifurcation-type theorem summarizes the situation for problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>.

Theorem 3.8If hypotheses H hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5">View MathML</a>such that

(a) for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6">View MathML</a>problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>has at least two positive solutions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M423">View MathML</a>

(b) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7">View MathML</a>problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M426">View MathML</a>;

(c) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8">View MathML</a>problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M15">View MathML</a>has no positive solution.

Remark 3.9 As the referee pointed out, it is an interesting problem to produce an example in which, at the bifurcation point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M429">View MathML</a>, the equation has exactly one solution. We believe that the recent paper of Gasiński and Papageorgiou [21] on the existence and uniqueness of positive solutions will be helpful. Concerning the existence of nodal solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M430">View MathML</a>, we mention the recent paper of Gasiński and Papageorgiou [22], which studies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1">View MathML</a>-equations and produces nodal solutions for them.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

The authors would like to express their gratitude to both knowledgeable referees for their corrections and remarks. This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.

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