Open Access Research

Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms

Tsing-San Hsu

Author Affiliations

Center for General Education, Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan ROC

Boundary Value Problems 2011, 2011:37  doi:10.1186/1687-2770-2011-37


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/37


Received:13 April 2011
Accepted:19 October 2011
Published:19 October 2011

© 2011 Hsu; 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

In this paper, we are concerned with the following quasilinear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M1">View MathML</a>

where Ω ⊂ ℝN is a smooth domain with smooth boundary ∂Ω such that 0 ∈ Ω, Δpu = div(|∇u|p-2u), 1 < p < N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2">View MathML</a>, λ >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M4">View MathML</a> is the best Hardy constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5">View MathML</a> is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solutions to this equation is verified.

Keywords:
Multiple positive solutions; critical Sobolev exponent; concave-convex; Hardy terms; sign-changing weights

1 Introduction and main results

Let Ω be a smooth domain (not necessarily bounded) in ℝN (N ≥ 3) with smooth boundary ∂Ω such that 0 ∈ Ω. We will study the multiplicity of positive solutions for the following quasilinear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M6">View MathML</a>

(1.1)

where Δpu = div(|∇u|p-2u), 1 < p < N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M7">View MathML</a> is the best Hardy constant, λ > 0, 1 < q < p, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M5">View MathML</a> is the critical Sobolev exponent and the weight functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M8">View MathML</a> are continuous, which change sign on Ω.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M10">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M11">View MathML</a>. The energy functional of (1.1) is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M12">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M13">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M14">View MathML</a> is said to be a solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M15">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M16">View MathML</a> and a solution of (1.1) is a critical point of Jλ.

Problem (1.1) is related to the well-known Hardy inequality [1,2]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M17">View MathML</a>

By the Hardy inequality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> has the equivalent norm ||u||μ, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M18">View MathML</a>

Therefore, for 1 < p < N, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19">View MathML</a>, we can define the best Sobolev constant:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M20">View MathML</a>

(1.2)

It is well known that Sμ(Ω) = Sμ(ℝN) = Sμ. Note that Sμ = S0 when μ ≤ 0 [3].

Such kind of problem with critical exponents and nonnegative weight functions has been extensively studied by many authors. We refer, e.g., in bounded domains and for p = 2 to [4-6] and for p > 1 to [7-11], while in ℝN and for p = 2 to [12,13], and for p > 1 to [3,14-17], and the references therein.

In the present paper, our research is mainly related to (1.1) with 1 < q < p < N, the critical exponent and weight functions f, g that change sign on Ω. When p = 2, 1 < q < 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M21">View MathML</a>, f, g are sign changing and Ω is bounded, [18] studied (1.1) and obtained that there exists Λ > 0 such that (1.1) has at least two positive solutions for all λ ∈ (0, Λ). For the case p ≠ 2, [19] studied (1.1) and obtained the multiplicity of positive solutions when 1 < q < p < N, μ = 0, f, g are sign changing and Ω is bounded. However, little has been done for this type of problem (1.1). Recently, Wang et al. [11] have studied (1.1) in a bounded domain Ω under the assumptions 1 < q < p < N, N > p2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M22">View MathML</a> and f, g are nonnegative. They also proved that there existence of Λ0 > 0 such that for λ ∈ (0, Λ0), (1.1) possesses at least two positive solutions. In this paper, we study (1.1) and extend the results of [11,18,19] to the more general case 1 < q < p < N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M22">View MathML</a>, f, g are sign changing and Ω is a smooth domain (not necessarily bounded) in ℝN (N ≥ 3). By extracting the Palais-Smale sequence in the Nehari manifold, the existence of at least two positive solutions of (1.1) is verified.

The following assumptions are used in this paper:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19">View MathML</a>, λ > 0, 1 < q < p < N, N ≥ 3.

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M24">View MathML</a>f+ = max{f, 0} ≢ 0 in Ω.

(f2) There exist β0 and ρ0 > 0 such that B(x0; 2ρ0) ⊂ Ω and f (x) ≥ β0 for all x B(x0; 2ρ0)

(g1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M25">View MathML</a> and g+ = max{g, 0} ≢ 0 in Ω.

(g2) There exist x0 ∈ Ω and β > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M26">View MathML</a>

where | · |denotes the L(Ω) norm.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M27">View MathML</a>

(1.3)

The main results of this paper are concluded in the following theorems. When Ω is an unbounded domain, the conclusions are new to the best of our knowledge.

Theorem 1.1 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>, (f1) and (g1) hold. Then, (1.1) has at least one positive solution for all λ ∈ (0, Λ1).

Theorem 1.2 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>, (f1) - (g2) hold, and γ is the constant defined as in Lemma 2.2. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28">View MathML</a>, x0 = 0 and β pγ, then (1.1) has at least two positive solutions for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>.

Theorem 1.3 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>, (f1) - (g2) hold. If μ < 0, x0 ≠ 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M30">View MathML</a>and N p2, then (1.1) has at least two positive solutions for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M31">View MathML</a>.

Remark 1.4 As Ω is a bounded smooth domain and p = 2, the results of Theorems 1.1, 1.2 are improvements of the main results of [18] .

Remark 1.5 As Ω is a bounded smooth domain and p ≠ 2, μ = 0, then the results of Theorems 1.1, 1.2 in this case are the same as the known results in [19] .

Remark 1.6 In this remark, we consider that Ω is a bounded domain. In [11] , Wang et al. considered (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19">View MathML</a>, λ > 0 and 1 < q < p < p2 < N. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28">View MathML</a>and 1 w< q < p < N, the results of Theorems 1.1, 1.2 are improvements of the main results of [11]. As μ < 0 and 1 < q < p < N p2, Theorem 1.3 is the complement to the results in [[11], Theorem 1.3].

This paper is organized as follows. Some preliminaries and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1-1.3 are proved in Sections 4-6, respectively. Before ending this section, we explain some notations employed in this paper. In the following argument, we always employ C and Ci to denote various positive constants and omit dx in integral for convenience. B(x0; R) is the ball centered at x0 ∈ ℝN with the radius R > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32">View MathML</a> denotes the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>, the norm in Lp(Ω) is denoted by |·|p, the quantity O(εt) denotes |O(εt)/εt| ≤ C, o(εt) means |o(εt)/εt| → 0 as ε → 0 and o(1) is a generic infinitesimal value. In particular, the quantity O1(εt) means that there exist C1, C2 > 0 such that C1εt O1(εt) ≤ C2εt as ε is small enough.

2 Preliminaries

Throughout this paper, (f1) and (g1) will be assumed. In this section, we will establish several preliminary lemmas. To this end, we first recall a result on the extremal functions of Sμ,s.

Lemma 2.1 [16]Assume that 1 < p < N and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28">View MathML</a>. Then, the limiting problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M33">View MathML</a>

(2.1)

has positive radial ground states

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M34">View MathML</a>

that satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M35">View MathML</a>

Furthermore, Up,μ(|x|) = Up,μ(r) is decreasing and has the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M36">View MathML</a>

where ci (i = 1, 2, 3, 4) are positive constants depending on N, μ and p, and a(μ) and b(μ) are the zeros of the function h(t) = (p - 1)tp - (N - p)tp-1 + μ, t ≥ 0, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M37">View MathML</a>.

Take ρ > 0 small enough such that B(0; ρ) ⊂ Ω, and define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M38">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M39">View MathML</a> is a cutoff function such that η(x) ≡ 1 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M40">View MathML</a>.

Lemma 2.2 [9,20]Suppose 1 < p < N and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M28">View MathML</a>. Then, the following estimates hold when ε → 0.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M41">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M43">View MathML</a>and γ = b(μ) - δ.

We also recall the following known result by Ben-Naoum, Troestler and Willem, which will be employed for the energy functional.

Lemma 2.3 [21]Let Ω be an domain, not necessarily bounded, in N, 1 ≤ p < N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M44">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M45">View MathML</a>Then, the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M46">View MathML</a>

is well-defined and weakly continuous.

3 Nehari manifold

As Jλ is not bounded below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>, we need to study Jλ on the Nehari manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M47">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a> contains all solutions of (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M50">View MathML</a>

(3.1)

Lemma 3.1 Jλ is coercive and bounded below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a>.

Proof Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49">View MathML</a>. From (f1), (3.1), the Hölder inequality and Sobolev embedding theorem, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M51">View MathML</a>

(3.2)

Thus, Jλ is coercive and bounded below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a>. □

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M52">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M53">View MathML</a>

(3.3)

Arguing as in [22], we split <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a> into three parts:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M54">View MathML</a>

Lemma 3.2 Suppose uλ is a local minimizer of Jλ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M55">View MathML</a>.

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M56">View MathML</a>in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32">View MathML</a>.

Proof The proof is similar to [[23], Theorem 2.3] and is omitted. □

Lemma 3.3 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M57">View MathML</a>for all λ ∈ (0, Λ1).

Proof We argue by contradiction. Suppose that there exists λ ∈ (0, Λ1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M202">View MathML</a>. Then, the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M58">View MathML</a> and (3.3) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M59">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M60">View MathML</a>

By (f1), (g1), the Hölder inequality and Sobolev embedding theorem, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M62">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M63">View MathML</a>

which is a contradiction. □

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M66">View MathML</a>

Lemma 3.4 Suppose that λ ∈ (0, Λ1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64">View MathML</a>is a function satisfying with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M67">View MathML</a> , then there exists a unique t- > tmax such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M68">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M69">View MathML</a>

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M67">View MathML</a>, then there exists a unique t± such that 0 < t+ < tmax < t-, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M70">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M68">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M71">View MathML</a>

Proof See Brown-Wu [[24], Lemma 2.6]. □

We remark that it follows Lemma 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M72">View MathML</a> for all λ ∈ (0, Λ1). Furthermore, by Lemma 3.4, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74">View MathML</a> are nonempty, and by Lemma 3.1, we may define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M75">View MathML</a>

Lemma 3.5 (i) If λ ∈ (0, Λ1), then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M76">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M77">View MathML</a>for some positive constant d0.

In particular, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M78">View MathML</a>.

Proof (i) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M79">View MathML</a>. From (3.3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M80">View MathML</a>

(3.4)

According to (3.1) and (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M81">View MathML</a>

By the definitions of αλ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M82">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M76">View MathML</a>.

(ii) Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M83">View MathML</a>. Then, (3.3) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M84">View MathML</a>

(3.5)

Moreover, by (g1) and the Sobolev embedding theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M85">View MathML</a>

(3.6)

From (3.5) and (3.6), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M86">View MathML</a>

(3.7)

By (3.2) and (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M87">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M88">View MathML</a>

for some positive constant d0. □

Remark 3.6 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M89">View MathML</a>, then by Lemmas 3.4 and 3.5, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M64">View MathML</a>with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M65">View MathML</a>, we can easily deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M90">View MathML</a>

4 Proof of Theorem 1.1

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values and (PS)-conditions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> for Jλ as follows:

Definition 4.1 (i) For c ∈ ℝ, a sequence {un} is a (PS)c-sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>for Jλ if Jλ(un) = c + o(1) and (Jλ)'(un) = o(1) strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M32">View MathML</a>as n → ∞.

(ii) c ∈ ℝ is a (PS)-value in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>for Jλ if there exists a (PS)c-sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>for Jλ.

(iii) Jλ satisfies the (PS)c-condition in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>if any (PS)c-sequence {un} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>for Jλ contains a convergent subsequence.

Lemma 4.2 (i) If λ ∈ (0, Λ1), then Jλ has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>, then Jλ has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M93">View MathML</a>.

Proof The proof is similar to [19,25] and the details are omitted. □

Now, we establish the existence of a local minimum for Jλ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a>.

Theorem 4.3 Suppose that N ≥ 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19">View MathML</a>, 1 < q < p < N and the conditions (f1), (g1) hold. If λ ∈ (0, Λ1), then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94">View MathML</a> such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M95">View MathML</a> ,

(ii) uλ is a positive solution of (1.1),

(iii) ||uλ||μ → 0 as λ → 0+.

Proof By Lemma 4.2 (i), there exists a minimizing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M96">View MathML</a>

(4.1)

Since Jλ is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a> (see Lemma 2.1), we get that (un) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>. Passing to a subsequence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M97">View MathML</a> such that as n → ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M98">View MathML</a>

(4.2)

By (f1) and Lemma 2.3, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M99">View MathML</a>

(3)

From (4.1)-(4.3), a standard argument shows that uλ is a critical point of Jλ. Furthermore, the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M92">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M100">View MathML</a>

(4.4)

Taking n → ∞ in (4.4), by (4.1), (4.3) and the fact αλ < 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M101">View MathML</a>

(4.5)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M102">View MathML</a> is a nontrivial solution of (1.1).

Next, we prove that un uλ strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> and Jλ(uλ) = αλ. From (4.3), the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M103">View MathML</a> and the Fatou's lemma it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M104">View MathML</a>

which implies that Jλ(uλ) = αλ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M105">View MathML</a>. Standard argument shows that un uλ strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M106">View MathML</a>, by Lemma 3.4, there exist unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M111">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M112">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M114">View MathML</a>. By Lemma 3.4, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M115">View MathML</a>

which is a contradiction. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M116">View MathML</a>, and by Jλ(uλ) = Jλ(|uλ|) = αλ, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M117">View MathML</a> is a local minimum of Jλ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M48">View MathML</a>. Then, by Lemma 3.2, we may assume that uλ is a nontrivial nonnegative solution of (1.1). By Harnack inequality due to Trudinger [26], we obtain that uλ > 0 in Ω. Finally, by (3.3), the Hölder inequality and Sobolev embedding theorem, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M118">View MathML</a>

which implies that ||uλ||μ → 0 as λ → 0+. □

Proof of Theorem 1.1 From Theorem 4.3, it follows that the problem (1.1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94">View MathML</a> for all λ ∈ (0, Λ0). □

5 Proof of Theorem 1.2

For 1 < p < N and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M19">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M119">View MathML</a>

Lemma 5.1 Suppose {un} is a bounded sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>. If {un} is a (PS)c-sequence for Jλ with c ∈ (0, c*), then there exists a subsequence of {un} converging weakly to a nonzero solution of (1.1).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M120">View MathML</a> be a (PS)c-sequence for Jλ with c ∈ (0, c*). Since {un} is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>, passing to a subsequence if necessary, we may assume that as n → ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M121">View MathML</a>

(5.1)

By (f1), (g1), (5.1) and Lemma 2.3, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M122">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M123">View MathML</a>

(5.2)

Next, we verify that u0 ≢ 0. Arguing by contradiction, we assume u0 ≡ 0. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M124">View MathML</a> as n → ∞ and {un} is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>, then by (5.2), we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M125">View MathML</a>

Then, we can set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M126">View MathML</a>

(5.3)

If l = 0, then we get c = limn→∞ Jλ(un) = 0, which is a contradiction. Thus, we conclude that l > 0. Furthermore, the Sobolev embedding theorem implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M127">View MathML</a>

Then, as n → ∞ we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M128">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M129">View MathML</a>

(5.4)

Hence, from (5.2)-(5.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M130">View MathML</a>

This is contrary to c < c*. Therefore, u0 is a nontrivial solution of (1.1). □

Lemma 5.2 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>and (f1) - (g2) hold. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M131">View MathML</a>, x0 = 0 and β pγ, then for any λ > 0, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M132">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M133">View MathML</a>

(5.5)

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M134">View MathML</a>for all λ ∈ (0, Λ1).

Proof From [[11], Lemma 5.3], we get that if ε is small enough, there exist tε > 0 and the positive constants Ci (i = 1, 2) independent of ε, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M135">View MathML</a>

(5.6)

By (g2), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M136">View MathML</a>

which together with Lemma 2.2 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M137">View MathML</a>

(5.7)

From the fact λ > 0, 1 < q < p, β and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M138">View MathML</a>

and by Lemma 2.2, (5.7) and (f2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M139">View MathML</a>

(5.8)

By (5.6) and (5.8), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M140">View MathML</a>

(5.9)

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M141">View MathML</a>, then by Lemma 2.2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M142">View MathML</a> we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M143">View MathML</a>

Combining this with (5.9), for any λ > 0, we can choose ελ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M144">View MathML</a>

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M145">View MathML</a>, then by Lemma 2.2 and γ > 0 we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M146">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M147">View MathML</a>

Combining this with (5.9), for any λ > 0, we can choose ελ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M148">View MathML</a>

From (i) and (ii), (5.5) holds by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M149">View MathML</a>.

In fact, by (f2), (g2) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M150">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M151">View MathML</a>

From Lemma 3.4, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M152">View MathML</a> and (5.5), for any λ ∈ (0, Λ0), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M153">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M154">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M155">View MathML</a>

The proof is thus complete. □

Now, we establish the existence of a local minimum of Jλ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74">View MathML</a>.

Theorem 5.3 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>and (f1) - (g2) hold. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M131">View MathML</a>, x0 = 0, β pγ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M157">View MathML</a>,

(ii) Uλ is a positive solution of (1.1).

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>, then by Lemmas 3.5 (ii), 4.2 (ii) and 5.2, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M91">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M93">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> for Jλ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M158">View MathML</a>. Since Jλ is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M74">View MathML</a> (see Lemma 3.1), we get that {un} is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>. From Lemma 5.1, there exists a subsequence still denoted by {un} and a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M159">View MathML</a> of (1.1) such that un Uλ weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>.

First, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156">View MathML</a>. On the contrary, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M160">View MathML</a>, then by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M161">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a>, we have ||Uλ||μ < lim infn→∞ ||un||μ. From (g2) and Uλ ≢ 0 in Ω, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M162">View MathML</a>. Thus, by Lemma 3.4, there exists a unique tλ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M163">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M49">View MathML</a>, then it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M164">View MathML</a>

(5.10)

From Remark 3.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M165">View MathML</a> and (5.10), we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M166">View MathML</a>

This is a contradiction. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M156">View MathML</a>.

Next, by the same argument as that in Theorem 4.3, we get that un Uλ strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M167">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M29">View MathML</a>. Since Jλ(Uλ) = Jλ(|Uλ|) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M168">View MathML</a>, by Lemma 3.2, we may assume that Uλ is a nontrivial nonnegative solution of (1.1). Finally, by Harnack inequality due to Trudinger [26], we obtain that Uλ is a positive solution of (1.1). □

Proof of Theorem 1.2 From Theorem 4.3, we get the first positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M94">View MathML</a> for all λ ∈ (0, Λ0). From Theorem 5.3, we get the second positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M160">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M89">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M169">View MathML</a>, this implies that uλ and Uλ are distinct. □

6 Proof of Theorem 1.3

In this section, we consider the case μ ≤ 0. In this case, it is well-known Sμ = S0 where Sμ is defined as in (1.2). Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M170">View MathML</a> when μ ≤ 0.

Lemma 6.1 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>and (f1) - (g2) hold. If N p2, μ < 0, x0 ≠ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M171">View MathML</a>, then for any λ > 0 and μ < 0, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M172">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M173">View MathML</a>

(6.1)

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M174">View MathML</a>for all λ ∈ (0, Λ1).

Proof Note that S0 has the following explicit extremals [27]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M175">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M176">View MathML</a> is a particular constant. Take ρ > 0 small enough such that B(x0; ρ) ⊂ Ω\{0} and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M177">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M178">View MathML</a> is a cutoff function such that φ(x) ≡ 1 in B(x0; ρ/2). Arguing as in Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M179">View MathML</a>

(6.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M180">View MathML</a>

(6.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M181">View MathML</a>

(6.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M182">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M184">View MathML</a>. Arguing as in Lemma 5.2, we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M185">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M186">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M187">View MathML</a>

(6.5)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M23">View MathML</a>, N p2 and (6.4), we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M188">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M189">View MathML</a>

Combining this with (6.5), for any λ > 0 and μ < 0, we can choose ελ,μ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M190">View MathML</a>

Therefore, (6.1) holds by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M191">View MathML</a>.

In fact, by (f2), (g2) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M192">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M193">View MathML</a>

From Lemma 3.4, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M152">View MathML</a> and (6.1), for any λ ∈ (0, Λ0) and μ < 0, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M194">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M195">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M196">View MathML</a>

The proof is thus complete. □

Proof of Theorem 1.3 Let Λ1(0) be defined as in (1.3). Arguing as in Theorems 4.3 and 5.3, we can get the first positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M197">View MathML</a> for all λ ∈ (0, Λ1(0)) and the second positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M198">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M31">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M199">View MathML</a>, this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/37/mathml/M201">View MathML</a> are distinct. □

Acknowledgements

The author is grateful for the referee's valuable suggestions.

References

  1. Hardy, G, Littlewood, J, Polya, G: Inequalities, Reprint of the 1952 edition, Cambridge Math. Lib. Cambridge University Press, Cambridge (1988)

  2. Caffarelli, L, Kohn, R, Nirenberg, L: First order interpolation inequality with weights. Compos Math. 53, 259–275 (1984)

  3. Abdellaoui, B, Felli, V, Peral, I: Existence and non-existence results for quasilinear elliptic equations involving the p-Laplacian. Boll Unione Mat Ital Scz B. 9, 445–484 (2006)

  4. Cao, D, Han, P: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differ Equ. 205, 521–537 (2004). Publisher Full Text OpenURL

  5. Ghoussoub, N, Robert, F: The effect of curvature on the best constant in the Hardy-Sobolev inequalities. Geom Funct Anal. 16, 897–908 (2006)

  6. Kang, D, Peng, S: Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl Math Lett. 18, 1094–1100 (2005). Publisher Full Text OpenURL

  7. Cao, D, Kang, D: Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J Math Anal Appl. 333, 889–903 (2007). Publisher Full Text OpenURL

  8. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Am Math Soc. 352, 5703–5743 (2000). Publisher Full Text OpenURL

  9. Han, P: Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal. 61, 735–758 (2005). Publisher Full Text OpenURL

  10. Kang, D: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 68, 1973–1985 (2008). Publisher Full Text OpenURL

  11. Wang, L, Wei, Q, Kang, D: Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardytype term. Nonlinear Anal. 74, 626–638 (2011). Publisher Full Text OpenURL

  12. Felli, V, Terracini, S: Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Commun Part Differ Equ. 31, 469–495 (2006). Publisher Full Text OpenURL

  13. Li, J: Existence of solution for a singular elliptic equation with critical Sobolev-Hardy exponents. Intern J Math Math Sci. 20, 3213–3223 (2005)

  14. Filippucci, R, Pucci, P, Robert, F: On a p-Laplace equation with multiple critical nonlinearities. J Math Pures Appl. 91, 156–177 (2009). Publisher Full Text OpenURL

  15. Gazzini, M, Musina, R: On a Sobolev-type inequality related to the weighted p-Laplace operator. J Math Anal Appl. 352, 99–111 (2009). Publisher Full Text OpenURL

  16. Kang, D: Solution of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy term. J Math Anal Appl. 341, 764–782 (2008). Publisher Full Text OpenURL

  17. Pucci, P, Servadei, R: Existence, non-existence and regularity of radial ground states for p-Laplace weights. Ann Inst H Poincaré Anal Non Linéaire. 25, 505–537 (2008). Publisher Full Text OpenURL

  18. Hsu, TS, Lin, HL: Multiple positive solutions for singular elliptic equations with concave-convex nonlinearities and sign-changing weights. Boundary Value Problems. 2009, 17 Article ID 584203) (2009)

  19. Hsu, TS: Multiplicity results for p-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. Abstr Appl Anal. 2009, 24 Article ID 652109) (2009)

  20. Kang, D, Huang, Y, Liu, S: Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J S Cent Univ Natl. 27, 91–95 (2008)

  21. Ben-Naoum, AK, Troestler, C, Willem, M: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 26, 823–833 (1996). Publisher Full Text OpenURL

  22. Tarantello, G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann Inst H Poincaré Anal Non Linéaire. 9, 281–304 (1992)

  23. Brown, KJ, Zhang, Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weigh function. J Differ Equ. 193, 481–499 (2003). Publisher Full Text OpenURL

  24. Brown, KJ, Wu, TF: A semilinear elliptic system involving nonlinear boundary condition and sign-changing weigh function. J Math Anal Appl. 337, 1326–1336 (2008). Publisher Full Text OpenURL

  25. Wu, TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J Math Anal Appl. 318, 253–270 (2006). Publisher Full Text OpenURL

  26. Trudinger, NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun Pure Appl Math. 20, 721–747 (1967). Publisher Full Text OpenURL

  27. Talenti, G: Best constant in Sobolev inequality. Ann Mat Pura Appl. 110, 353–372 (1976). Publisher Full Text OpenURL