Open Access Research

Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN

Huei-li Lin

Author affiliations

Department of Natural Sciences in the Center for General Education, Chang Gung University, Tao-Yuan 333, Taiwan

Citation and License

Boundary Value Problems 2012, 2012:24  doi:10.1186/1687-2770-2012-24


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/24


Received:13 July 2011
Accepted:24 February 2012
Published:24 February 2012

© 2012 Lin; 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 article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear elliptic equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M1">View MathML</a>

where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3.

AMS (MOS) subject classification: 35J20; 35J25; 35J65.

Keywords:
semilinear elliptic equations; concave and convex; positive solutions

1 Introduction

For N ≥ 3, 1 ≤ q < 2 < p < 2* = 2N/(N - 2), we consider the semilinear elliptic equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M230">View MathML</a>

where λ > 0.

Let f and h satisfy the following conditions:

(f 1) f is a positive continuous function in ℝN and lim|z| → ∞ f(z) = f> 0.

(f2) there exist k points a1, a2,..., ak in ℝN such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M2">View MathML</a>

and f< fmax.

(h 1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M4">View MathML</a>.

Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M5">View MathML</a>

have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2]<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M6">View MathML</a> and changes sign, 1 < q < 2 < p < 2*). They proved that this equation has at least two positive solutions for sufficiently small c > 0. More general results of Equation (Ec) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].

In this article, we consider the existence and multiplicity of positive solutions of Equation (Eλ) in ℝN. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝN, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (Eλ) admits at least two positive solutions in ℝN. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (Eλ) admits at least two positive solutions in ℝN. For the case q = λ = 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [9] asserted that Equation (Eλ) admits at least four positive solutions in ℝN, where f(z) ≢ 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M7">View MathML</a>. Similarly, in Hsu and Lin [10], they have studied that there are at least four positive solutions of the general case -Δu + u = f(z)vp-1 + λh(z) vq-1 in ℝN for sufficiently small λ > 0.

By the change of variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M8">View MathML</a>

Equation (Eλ) is transformed to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M9">View MathML</a>

Associated with Equation (Eε), we consider the C1-functional Jε, for u H1 (ℝN),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M11">View MathML</a> is the norm in H1 (ℝN) and u+ = max{u, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (Eε) are equivalent to the critical points of Jε. This article is organized as follows. First of all, we use the argument of Tarantello [11] to divide the Nehari manifold Mε into the two parts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M13">View MathML</a>. Next, we prove that the existence of a positive ground state solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M14">View MathML</a> of Equation (Eε). Finally, in Section 4, we show that the condition (f2) affects the number of positive solutions of Equation (Eε), that is, there are at least k critical points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M15">View MathML</a> of Jε such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M16">View MathML</a> for 1 ≤ i k.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M17">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M18">View MathML</a>

(1.1)

For the semilinear elliptic equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M19">View MathML</a>

(E0)

we define the energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M20">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M21">View MathML</a>

where Nε = {u H1 (ℝN) \ {0} | u+ ≢ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M22">View MathML</a>}. Note that

(i) if f f, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M23">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M24">View MathML</a>

where N= {u H1 (ℝN) \ {0} | u+ ≢ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M25">View MathML</a>};

(ii) if f fmax, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M26">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M27">View MathML</a>

where Nmax = {u H1 (ℝN) \ {0} | u+ ≢ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M28">View MathML</a>}.

Lemma 1.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M29">View MathML</a>

Proof. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].

Our main results are as follows.

(I) Let Λ = ε2(p-q)/(p-2). Under assumptions (f 1) and (h1), if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M30">View MathML</a>

where ∥h# is the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M31">View MathML</a>, then Equation (Eε) admits at least a positive ground state solution. (See Theorem 3.4)

(II) Under assumptions (f1) - (f2) and (h1), if λ is sufficiently large, then Equation (Eλ) admits at least k + 1 positive solutions. (See Theorem 4.8)

2 The Nehari manifold

First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H1(ℝN) for some functional J.

Definition 2.1 (i) For β ∈ ℝ, a sequence {un} is a (PS)β-sequence in H1(ℝN) for J if J(un) = β + on(1) and J'(un) = on(1) strongly in H-1 (ℝN) as n → ∞, where H-1 (ℝN) is the dual space of H1(ℝN);

(ii) J satisfies the (PS)β-condition in H1(ℝN) if every (PS)β-sequence in H1(ℝN) for J contains a convergent subsequence.

Next, since Jε is not bounded from below in H1 (ℝN), we consider the Nehari manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M32">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M33">View MathML</a>

Note that Mε contains all nonnegative solutions of Equation (Eε). From the lemma below, we have that Jε is bounded from below on Mε.

Lemma 2.2 The energy functional Jε is coercive and bounded from below on Mε.

Proof. For u Mε, by (2.1), the Hölder inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M34">View MathML</a> and the Sobolev embedding theorem (1.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M35">View MathML</a>

Hence, we have that Jε is coercive and bounded from below on Mε.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M36">View MathML</a>

Then for u Mε, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M37">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M38">View MathML</a>

(2.3)

We apply the method in Tarantello [11], let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M39">View MathML</a>

Lemma 2.3 Under assumptions (f1) and (h1), if 0 < Λ (= ε2(p-q)/(p-2)) < Λ0, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M40">View MathML</a>.

Proof. See Hsu and Lin [[10], Lemma 5].

Lemma 2.4 Suppose that u is a local minimizer for Jε on Mε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M41">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M42">View MathML</a>in H-1 (ℝN).

Proof. See Brown and Zhang [[4], Theorem 2.3].

Lemma 2.5 We have the following inequalities.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M43">View MathML</a>for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M45">View MathML</a>for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M46">View MathML</a>for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47">View MathML</a>;

(iv) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M48">View MathML</a>, then Jε(u) > 0 for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47">View MathML</a>.

Proof. (i) It can be proved by using (2.2).

(ii) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M49">View MathML</a>, by (2.2), we apply the Hölder inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M50">View MathML</a> to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M51">View MathML</a>

(iii) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M47">View MathML</a>, by (2.3), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M52">View MathML</a>

(iv) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M53">View MathML</a>, by (iii), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M54">View MathML</a>

Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M55">View MathML</a>, we get that Jε(u) ≥ d0 > 0 for some constant d0 = d0(ε, p, q, S, ∥h# , fmax).

For u H1 (ℝN) \ {0} and u+ ≢ 0, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M56">View MathML</a>

Lemma 2.6 For each u H1 (ℝN)\ {0} and u+ ≢ 0, we have that

(i) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M57">View MathML</a>, then there exists a unique positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M58">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M59">View MathML</a>and Jε(t-u) = supt ≥ 0 Jε(tu);

(ii) if 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M60">View MathML</a>, then there exist unique positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M61">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M62">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M63">View MathML</a>

Proof. See Hsu and Lin [[10], Lemma 7].

Applying Lemma 2.3 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M64">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M65">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M66">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M67">View MathML</a>

Lemma 2.7 (i) If 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M68">View MathML</a>;

(ii) If 0 < Λ < qΛ0/2, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M69">View MathML</a>for some constant d0 = d0 (ε, p, q, S, ∥h#, fmax).

Proof. (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M44">View MathML</a>, by (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M70">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M71">View MathML</a>

By the definitions of αε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M72">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M73">View MathML</a>.

(ii) See the proof of Lemma 2.5 (iv).

Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.

Lemma 2.8 (i) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M74">View MathML</a>-sequence {un} in Mε for Jε;

(ii) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M75">View MathML</a>-sequence {un} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M76">View MathML</a>for Jε;

(iii) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M77">View MathML</a>-sequence {un} in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M78">View MathML</a>for Jε.

3 Existence of a ground state solution

In order to prove the existence of positive solutions, we claim that Jε satisfies the (PS)β-condition in H1(ℝN) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M79">View MathML</a>, where Λ = ε2(p-q)/(p-2) and C0 is defined in the following lemma.

Lemma 3.1 Assume that h satisfies (h1) and 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0. If {un} is a (PS)β-sequence in H1(ℝN) for Jε with un u weakly in H1 (ℝN), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80">View MathML</a>in H-1 (ℝN) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M81">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M82">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M83">View MathML</a>

Proof. Since {un} is a (PS)β-sequence in H1(ℝN) for Jε with un u weakly in H1 (ℝN), it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80">View MathML</a> in H-1(ℝN) and u ≥ 0. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M84">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M85">View MathML</a>. Hence, by the Young inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M86">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M87">View MathML</a>

Lemma 3.2 Assume that f and h satisfy (f1) and (h1). If 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then Jε satisfies the (PS)β-condition in H1(ℝN) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M79">View MathML</a>.

Proof. Let {un} be a (PS)β-sequence in H1(ℝN) for Jε such that Jε(un) = β + on(1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M88">View MathML</a> (1) in H-1(ℝN). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M89">View MathML</a>

where cn = on(1), dn = on(1) as n → ∞. It follows that {un} is bounded in H1(ℝN). Hence, there exist a subsequence {un} and a nonnegative u H1 (ℝN) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M80">View MathML</a> in H-1 (ℝN), un u weakly in H1 (ℝN), un u a.e. in ℝN, un u strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M90">View MathML</a> for any 1 ≤ s < 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M91">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M92">View MathML</a>

(3.2)

Next, claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M93">View MathML</a>

(3.3)

For any σ > 0, there exists r > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M94">View MathML</a>. By the Hölder inequality and the Sobolev embedding theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M95">View MathML</a>

Applying (f1) and un u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M96">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M97">View MathML</a>

(3.4)

Let pn = un - u. Suppose pn ↛ 0 strongly in H1 (ℝN). By (3.1)-(3.4), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M98">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M99">View MathML</a>

By Theorem 4.3 in Wang [12], there exists a sequence {sn} ⊂ ℝ+ such that sn = 1 + on(1), {sn pn} ⊂ Nand I(sn pn) = I(pn) + on(1). It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M100">View MathML</a>

which is a contradiction. Hence, un u strongly in H1(ℝN).

Remark 3.3 By Lemma 1.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M101">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M102">View MathML</a>for 0 < Λ < Λ0.

By Lemma 2.8 (i), there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M103">View MathML</a>-sequence {un} in Mε for Jε. Then we prove that Equation (Eε) admits a positive ground state solution u0 in ℝN.

Theorem 3.4 Under assumptions (f1), (h1), if 0 < Λ ( = ε2(p-q)/(p-2)) < Λ0, then there exists at least one positive ground state solution u0 of Equation (Eε) in N. Moreover, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M105">View MathML</a>

(3.5)

Proof. By Lemma 2.8 (i), there is a minimizing sequence {un} ⊂ Mε for Jε such that Jε(un) = αε + on(1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M106">View MathML</a> in H-1 (ℝN). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M107">View MathML</a>, by Lemma 3.2, there exist a subsequence {un} and u0 H1 (ℝN) such that un u0 strongly in H1 (ℝN). It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M108">View MathML</a> is a solution of Equation (Eε) in ℝN and Jε(u0) = αε. Next, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104">View MathML</a>. On the contrary, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M109">View MathML</a>.

We get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M110">View MathML</a>

Otherwise,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M111">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M112">View MathML</a>

which contradicts to αε < 0. By Lemma 2.6 (ii), there exist positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M114">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M115">View MathML</a>

which is a contradiction. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M104">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M116">View MathML</a>

By Lemma 2.4 and the maximum principle, then u0 is a positive solution of Equation (Eε) in ℝN.

4 Existence of k + 1 solutions

From now, we assume that f and h satisfy (f1)-(f2) and (h1). Let w H1 (ℝN) be the unique, radially symmetric, and positive ground state solution of Equation (E0) in ℝN for f = fmax. Recall the facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M117">View MathML</a> for some 0 < θ < 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M118">View MathML</a>;

(ii) for any ε > 0, there exist positive numbers C1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M119">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M120">View MathML</a> such that for all z ∈ ℝN

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M121">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M122">View MathML</a>

For 1 ≤ i k, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M123">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M124">View MathML</a>. By Lemma 2.6 (ii), there is a unique number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M126">View MathML</a>, where 1 ≤ i k.

We need to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M127">View MathML</a>

Lemma 4.1 (i) There exists a number t0 > 0 such that for 0 t t0 and any ε > 0, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M128">View MathML</a>

(ii) There exist positive numbers t1 and ε1 such that for any t > t1 and ε < ε1, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M129">View MathML</a>

Proof. (i) Since Jε is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M130">View MathML</a> is uniformly bounded in H1 (ℝN) for any ε > 0, and γmax > 0, there is t0 > 0 such that for 0 ≤ t t0 and any ε > 0

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M131">View MathML</a>

(ii) There is an r0 > 0 such that f (z) ≥ fmax/2 for z BN (ai; r0) uniformly in i. Then there exists ε1 > 0 such that for ε < ε1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M132">View MathML</a>

Thus, there is t1 >0 such that for any t > t1 and ε < ε1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M133">View MathML</a>

Lemma 4.2 Under assumptions (f1), (f2), and (h1). If 0 < Λ ( = ε2(p-q)/(p-2)) < q Λ0/2, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M134">View MathML</a>

Proof. By Lemma 4.1, we only need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M135">View MathML</a>

We know that supt ≥0 Imax (tw) = γmax. For t0 t t1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M136">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M137">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M138">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M139">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M140">View MathML</a> uniformly in i.

Applying the results of Lemmas 2.6, 2.7(ii), and 4.2, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M141">View MathML</a>

Since γmax < γ, there exists ε0 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M142">View MathML</a>

(4.1)

Choosing 0 < ρ0 < 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M143">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M144">View MathML</a> and f(ai) = fmax. Define K = {ai | 1 ≤ i k} and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M145">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M146">View MathML</a> for some r0 > 0.

Let Qε : H1 (ℝN) \ {0} → ℝN be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M147">View MathML</a>

where χ : ℝN → ℝN, χ (z) = z for |z| ≤ r0 and χ (z) = r0z/|z| for |z| > r0.

Lemma 4.3 There exists 0 < ε0 ε0 such that if ε < ε0, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M148">View MathML</a>for each 1 ≤ i k.

Proof. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M149">View MathML</a>

there exists ε0 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M150">View MathML</a>

Lemma 4.4 There exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M151">View MathML</a>such that if u Nε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M152">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153">View MathML</a>for any 0 < ε < ε0.

Proof. On the contrary, there exist the sequences {εn} ⊂ ℝ+ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M155">View MathML</a> (1) as n → ∞ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M156">View MathML</a> for all n ∈ ℕ. It is easy to check that {un} is bounded in H1 (ℝN). Suppose un → 0 strongly in Lp (ℝN). Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M157">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M158">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M159">View MathML</a>

which is a contradiction. Thus, un ↛ 0 strongly in Lp (ℝN). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d0 > 0 and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M161">View MathML</a>

(4.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M162">View MathML</a>, there are a subsequence {vn} and v H1 (ℝN) such that vn v weakly in H1 (ℝN). Using the similar computation in Lemma 2.6, there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M163">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M164">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M165">View MathML</a>

We deduce that a convergent subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M166">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M167">View MathML</a>. Then there are subsequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M170">View MathML</a> weakly in H1 (ℝN). By (4.2), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M171">View MathML</a>. Moreover, we can obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M172">View MathML</a> strongly in H1 (ℝN) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M173">View MathML</a>. Now, we want to show that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M174">View MathML</a> such that zn z0 K.

(i) Claim that the sequence {zn} is bounded in ℝN. On the contrary, assume that |zn| → ∞, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M175">View MathML</a>

which is a contradiction.

(ii) Claim that z0 K. On the contrary, assume that z0 K, that is, f(z0) < fmax. Then using the above argument to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M176">View MathML</a>

which is a contradiction. Since vn v ≠ 0 in H1 (ℝN), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M177">View MathML</a>

which is a contradiction.

Hence, there exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M151">View MathML</a> such that if u Nε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M152">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153">View MathML</a> for any 0 < ε < ε0.

From (4.1), choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M179">View MathML</a>

(4.3)

For each 1 ≤ i k, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M180">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M182">View MathML</a>.

Lemma 4.5 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M183">View MathML</a>and Jε (u) ≤ γmax + δ0/2, then there exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M184">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153">View MathML</a>for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185">View MathML</a>.

Proof. We use the similar computation in Lemma 2.6 to get that there is a unique positive number

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M186">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M187">View MathML</a>. We want to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M188">View MathML</a> for some constant c > 0 (independent of u). First, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M189">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M190">View MathML</a>

and Jε is coercive on Mε, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M191">View MathML</a> for some constants c1 and c2 (independent of u). Next, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M192">View MathML</a> for some constant c3 > 0 (independent of u). On the contrary, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M193">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M194">View MathML</a>

By (2.3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M195">View MathML</a>

which is a contradiction. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M188">View MathML</a> for some constant c > 0 (independent of u). Now, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M196">View MathML</a>

From the above inequality, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M197">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M184">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M198">View MathML</a>

By Lemma 4.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M199">View MathML</a>

or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M153">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M185">View MathML</a>.

Applying the above lemma, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M200">View MathML</a>

(4.4)

By Lemmas 4.2, 4.3, and Equation (4.3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M201">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M202">View MathML</a>

(4.5)

Lemma 4.6 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M203">View MathML</a>, then there exist an η > 0 and a differentiable functional l : B(0; η) ⊂ H1(ℝN) → ℝ+ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M204">View MathML</a>for any v B(0;η) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M205">View MathML</a>

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M206">View MathML</a>.

Proof. See Cao and Zhou [7].

Lemma 4.7 For each 1 ≤ i k, there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M207">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M208">View MathML</a>in H1(ℝN) for Jε.

Proof. For each 1 ≤ i k, by (4.4) and (4.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M209">View MathML</a>

(4.7)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M210">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M211">View MathML</a> be a minimizing sequence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M212">View MathML</a>. Applying Ekeland's variational principle, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M214">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M215">View MathML</a>

(4.8)

Using (4.7), we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M216">View MathML</a> for sufficiently large n. By Lemma 4.6, then there exist an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M217">View MathML</a> and a differentiable functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M218">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M219">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M220">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M221">View MathML</a>. Let vσ = σv with ║vH = 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M222">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M224">View MathML</a>. From (4.8) and by the mean value theorem, we get that as σ → 0

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M225">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M226">View MathML</a>

Since we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M227">View MathML</a> for all n and i from (4.6), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M228">View MathML</a> strongly in H-1 (ℝN) as n → ∞.

Theorem 4.8 Under assumptions (f1), (f2), and (h1), there exists a positive number λ*(λ* = (ε*)-2) such that for λ > λ*, Equation (Eλ) has k + 1 positive solutions in N.

Proof. We know that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M207">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M193">View MathML</a> in H1(ℝN) for Jε for each 1 ≤ i k, and (4.5). Since Jε satisfies the (PS)β-condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M229">View MathML</a>, then Jε has at least k critical points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/24/mathml/M13">View MathML</a> for 0 < ε < ε*. It follows that Equation (Eλ) has k nonnegative solutions in ℝN. Applying the maximum principle and Theorem 3.4, Equation (Eλ) has k + 1 positive solutions in ℝN.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author was grateful for the referee's helpful suggestions and comments.

References

  1. Ambrosetti, A, Brezis, H, Cerami, G: Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal. 122, 519–543 (1994). Publisher Full Text OpenURL

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

  3. Ambrosetti, A, Garcia Azorero, J, Peral Alonso, I: Multiplicity results for some nonlinear elliptic equations. J Funct Anal. 137, 219–242 (1996). Publisher Full Text OpenURL

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

  5. de Figueiredo, DG, Gossez, JP, Ubilla, P: Local superlinearity and sub-linearity for indefinite semilinear elliptic problems. J Funct Anal. 199, 452–467 (2003). Publisher Full Text OpenURL

  6. Zhu, XP: A perturbation result on positive entire solutions of a semilinear elliptic equation. J Diff Equ. 92, 163–178 (1991). Publisher Full Text OpenURL

  7. Cao, DM, Zhou, HS: Multiple positive solutions of nonhomogeneous semi-linear elliptic equations in ℝN. Proc Roy Soc Edinburgh, Sect A. 126, 443–463 (1996). Publisher Full Text OpenURL

  8. Hirano, N: Existence of entire positive solutions for nonhomogeneous elliptic equations. Nonlinear Anal. 29, 889–901 (1997). Publisher Full Text OpenURL

  9. Adachi, S, Tanaka, K: Four positive solutions for the semilinear elliptic equation: -Δu + u = a(x)up + f(x) in ℝN. Calc Var Partial Diff Equ. 11, 63–95 (2000). Publisher Full Text OpenURL

  10. Hsu, TS, Lin, HL: Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN. J Math Anal Appl. 365, 758–775 (2010). Publisher Full Text OpenURL

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

  12. Wang, HC: Palais-Smale approaches to semilinear elliptic equations in unbounded domains. Electron J Diff Equ Monogragh. 142 06 (2004)

  13. Bahri, A, Li, YY: On a min-max procedure for the existence of a positive solution for certain scalar field equations in ℝN. Rev Mat Iberoamericana. 6, 1–15 (1990)

  14. Bahri, A, Lions, PL: On the existence of a positive solution of semilin-ear elliptic equations in unbounded domains. Ann Inst H Poincaré Anal Nonlinéaire. 14, 365–413 (1997)

  15. Gidas, B, Ni, WM, Nirenberg, L: Symmetry and related properties via the maximum principle. Comm Math Phys. 68, 209–243 (1979). Publisher Full Text OpenURL

  16. Kwong, MK: Uniqueness of positive solutions of Δu - u + up = 0 in ℝN. Arch Ration Mech Anal. 105, 234–266 (1989)

  17. Lions, PL: The concentration-compactness principle in the calculus of variations. The locally compact case. I II Ann Inst H Poincaré Anal Non-linéaire. 1, 109–145 223-283 (1984)