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Multiple solutions for p-Laplacian systems with critical homogeneous nonlinearity

Dengfeng Lü

Author affiliations

School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, P. R. China

Citation and License

Boundary Value Problems 2012, 2012:27  doi:10.1186/1687-2770-2012-27


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/27


Received:17 November 2011
Accepted:28 February 2012
Published:28 February 2012

© 2012 Lü; 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 deal with existence and multiplicity of solutions to the p-Laplacian system of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1">View MathML</a>

where Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, Δpu = div(|∇u|p-2u) is the p-Laplacian operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2">View MathML</a> denotes the Sobolev critical exponent, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3">View MathML</a> is a homogeneous function of degree p*. By using the variational method and Ljusternik-Schnirelmann theory, we prove that the system has at least cat(Ω) distinct nonnegative solutions.

AMS 2010 Mathematics Subject Classifications: 35J50; 35B33.

Keywords:
p-Laplacian system; Ljusternik-Schnirelmann theory; critical exponent; multiple solutions

1 Introduction and main results

In this article, we consider the existence and multiplicity of solutions for the following critical p-Laplacian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M1">View MathML</a>

(1.1)

where Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, Δpu = div(|∇u|p-2u) is the p-Laplacian operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M2">View MathML</a> denotes the Sobolev critical exponent, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3">View MathML</a> is a homogeneous function of degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M4">View MathML</a> and λ, δ are positive parameters.

The starting point on the study of the system (1.1) is its scalar version:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M5">View MathML</a>

(1.2)

with 2 ≤ p q < p*. In a pioneer work Brezis and Nirenberg [1] showed that, if p = q = 2, the equation (1.2) has at least one positive solution provided N ≥ 4 and 0 < λ < λ1, where λ1 is the first eigenvalue of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M6">View MathML</a>. In particular, the first multiplicity result for (1.2) has been achieved by Rey [2] in the semilinear case. Precisely Rey proved that if N ≥ 5, p = q = 2, for λ small enough equation (1.2) has at least cat(Ω) solutions, where cat(Ω) denotes the Ljusternik-Schnirelmann category of Ω in itself. Furthermore, Alves and Ding [3] obtained the existence of cat(Ω) positive solutions to equation (1.2) with p ≥ 2, p q < p*.

In recent years, more and more attention have been paid to the elliptic systems. In particular, Ding and Xiao [4] concerned the case F(x, u, v) = 2|u|α|v|β,α > 1, β >1 satisfying α + β = p*, i.e., the following elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M7">View MathML</a>

(1.3)

Using standard tools of the variational theory and the Ljusternik-Schnirelmann category theory, Ding and Xiao [4] have proved that system (1.3) has at least cat(Ω) positive solutions if λ, δ satisfied a certain condition. Hsu [5] obtained the existence of two positive solutions of system (1.3) with the sublinear perturbation of 1 < q < p < N. Recently, Shen and Zhang [6] extended the results in [5] to the case (1.1) with 1 < q < p < N and obtained similar results. In this article, we study (1.1) and complement the results of [5,6] to the case 2 ≤ p q < p*, also extend the results of [4,7]. To the best of our knowledge, problem (1.1) has not been considered before. Thus it is necessary for us to investigate the critical p-Laplacian systems (1.1) deeply. For more similar problems, we refer to [8-17], and references therein.

Before stating our results, we need the following assumptions:

(F0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M8">View MathML</a> holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M9">View MathML</a>;

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M10">View MathML</a>, where u, v ∈ ℝ+;

(F2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M11">View MathML</a> are strictly increasing functions about u and v for all u, v > 0.

The main results we get are the following:

Theorem 1.1. Suppose N p2 and F satisfies (F0)-(F2), then the problem (1.1) has at least one nonnegative solution for 2 ≤ p < q < p* and λ, δ > 0, or q = p and λ, δ ∈ (0, Λ1), where Λ1 is the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12">View MathML</a>.

Theorem 1.2. Suppose N p2, 2 ≤ p q < p* and F satisfies (F0)-(F2), then there exists Λ > 0 such that the problem (1.1) has at least cat(Ω) distinct nonnegative solutions for λ, δ ∈ (0,Λ).

Remark 1.1. Theorem 1 in [4]is the special case of our Theorem 1.2 corresponding to F(x,u,v) = 2|u|α|v|β,α > 1,β > 1,α + β = p*. There are functions F(x,u,v) satisfying the conditions of our Theorems 1.1 and 1.2. Some typical examples are:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M13">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M15">View MathML</a>. Obviously, F(x, u, v) satisfies (F0)-(F2).

This article is organized as follows. In Section 2, some notations and the Mountain-Pass levels are established and the Theorem 1.1 is proved. We present some technical lemmas which are crucial in the proof of the Theorem 1.2 in Section 3. Theorem 1.2 is proved in Section 4.

2 Notations and proof of Theorem 1.1

Throughout this article, C, Ci will denote various positive constants whose exact values are not important, → (respectively ⇀) denotes strong (respectively weak) convergence. O(εt) denotes |O(εt)|/εt C, om(1) denotes om(1) → 0 as m → ∞. Ls(Ω)(1 ≤ s < +∞) denotes Lebesgue spaces, the norm Ls is denoted by | · |s for 1 ≤ s < + ∞. Let Br(x) denotes a ball centered at x with radius r, the dual space of a Banach space E will be denoted by E-1. We define the product space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M16">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M17">View MathML</a>, and the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M18">View MathML</a>.

Using assumption of (F1), we have the so-called Euler identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M19">View MathML</a>

(2.1)

In addition, we can extend the function F(x,u,v) to the whole <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M20">View MathML</a> by considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M21">View MathML</a>, where u+ = max{u,0}. It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M22">View MathML</a> is of class C1 and its restriction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M23">View MathML</a> coincides with F(x,u,v). In order to simplify the notation we shall write, from now on, only F(x,u,v) to denote the above extension.

A pair of functions (u, v) ∈ E is said to be a weak solution of problem (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M24">View MathML</a>

Thus, by (2.1) the corresponding energy functional of problem (1.1) is defined on E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M25">View MathML</a>

Using (F0)-(F2), we can verify Iλ, δ(u, v) ∈ C1(E, ℝ) (see [6]). It is well known that the weak solutions of problem (1.1) are the critical points of the energy functional Iλ, δ(u, v).

The functional I C1(E, ℝ) is said to satisfy the (PS)c condition if any sequence {um} ⊂ E such that as m → ∞, I(um) → c, I'(um) → 0 strongly in E-1 contains a subsequence converging in E to a critical point of I. In this article, we will take I = Iλ, δ(u, v) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M16">View MathML</a>.

As the energy functional Iλ,δ is not bounded below on E, we need to study Iλ,δ on the Nehari manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M26">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27">View MathML</a> contains every nonzero solution of problem (1.1), and define the minimax cλ,δ as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M28">View MathML</a>

Next, we present some properties of cλ,δ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27">View MathML</a>. Its proofs can be done as [18, Theorem 4.2]. First of all, we note that there exists ρ > 0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M29">View MathML</a>

It is standard to check that Iλ,δ satisfies Mountain-Pass geometry, so we can use the homogeneity of F to prove that cλ,δ can be alternatively characterized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M30">View MathML</a>

(2.2)

where Γ = {γ C([0, 1],E) : γ(0) = 0,Iλ,δ(γ(1)) < 0}. Moreover, for each (u, v) ∈ E\{(0,0)}, there exists a unique t* > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M31">View MathML</a>. The maximum of the function t Iλ,δ(t(u, v)), for t ≥ 0, is achieved at t = t*.

In this section, we will find the range of c where the (PS)c condition holds for the functional Iλ,δ. First let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M32">View MathML</a>

(2.3)

Lemma 2.1. If N p2 and F satisfies (F0)-(F2), then the functional Iλ,δ satisfies the (PS)c condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M33">View MathML</a>, provide one of the following conditions holds

(i) 2 ≤ p < q < p* and λ, δ > 0;

(ii) q = p, and λ, δ ∈ (0, Λ1), where Λ1 > 0 denotes the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12">View MathML</a>.

Proof. Let {(um, vm)} ⊂ E such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M35">View MathML</a>. Now, we first prove that {(um, vm)} is bounded in E. If the above item (i) is true it suffices to use the definition of Iλ,δ to obtain C1 > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M36">View MathML</a>

The above expression implies that {(um, vm)} ⊂ E is bounded. When (ii) occurs, in this case, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M37">View MathML</a>

and therefore we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M38">View MathML</a>

Since λ, δ ∈ (0,Λ1) the boundedness of {(um, vm)} follows as the first case.

So, {(um, vm)} is bounded in E. Going if necessary to a subsequence, we can assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M39">View MathML</a>

as m → ∞. Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M40">View MathML</a>

(2.4)

Moreover, a standard argument shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M41">View MathML</a>. Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M42">View MathML</a>

(2.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M43">View MathML</a>, then by Brezis-Lieb Lemma in [19] implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M44">View MathML</a>

(2.6)

By the same method of [8, Lemma 5] (or [6, Lemma 3.4]), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M45">View MathML</a>

(2.7)

By (2.4)-(2.7) and the weak convergence of (um, vm), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M46">View MathML</a>

(2.8)

By using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34">View MathML</a> and (2.4), (2.6), and (2.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M47">View MathML</a>

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M41">View MathML</a>, we can use the above equality and (2.8) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M48">View MathML</a>

where k is a nonnegative number.

In view of the definition of SF, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M49">View MathML</a>

Taking the limit we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M50">View MathML</a>. So, if k > 0, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M51">View MathML</a> and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M52">View MathML</a>

which is a contradiction. Hence k = 0 and therefore (um, vm) → (u, v) strongly in E.

Before presenting our next result we recall that, for each ε > 0, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M53">View MathML</a>

(2.9)

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M54">View MathML</a>

(2.10)

where S is the best constant of the Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M55">View MathML</a>. Thus, using [8, Lemma 3] and the homogeneity of F, we obtain A, B > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M56">View MathML</a>

from which and (2.10) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M57">View MathML</a>

(2.11)

We define a cut-off function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M58">View MathML</a> such that ϕ(x) = 1 if |x| ≤ R; ϕ(x) = 0 if |x| ≥ 2R and 0 ≤ ϕ(x) ≤ 1, where B2R(0) ⊂ Ω, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M59">View MathML</a>, where Uε was defined in (2.9). So that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M60">View MathML</a>. Then, we can get the following results from [[20], Lemma 11.1]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M61">View MathML</a>

(2.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M62">View MathML</a>

(2.13)

where A B means C1B A C2B.

Lemma 2.2. Suppose that F satisfies (F0)-(F2), 2 ≤ p < q < p* and λ > 0, δ > 0, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M63">View MathML</a>. The same result holds if q = p and λ, δ ∈ (0,Λ1), where Λ1 > 0 denotes the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M12">View MathML</a>.

Proof. We can use the homogeneity of F to get, for any t ≥ 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M64">View MathML</a>

We shall denote by h(t) the right-hand side of the above equality and consider two distinct cases.

Case 1. 2 ≤ p < q < p*.

From the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M65">View MathML</a> and h(t) > 0 when t is close to 0, there exists tε > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M66">View MathML</a>

(2.14)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M67">View MathML</a>

and notice that the maximum value of g(t) occurs at the point

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M68">View MathML</a>

So, for each t ≥ 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M69">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M70">View MathML</a>

(2.15)

We claim that, for some C2 > 0, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M71">View MathML</a>

Indeed, if this is not the case, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M72">View MathML</a> for some sequence εm → 0+, then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M73">View MathML</a>

which is a contradiction. So, the claim holds and we infer from (2.15) and (2.11)-(2.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M74">View MathML</a>

(2.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M75">View MathML</a>. We know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M76">View MathML</a> if N p2. By N p2 and 2 ≤ p < q < p* we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M77">View MathML</a>. Thus from the above inequality we conclude that, for each ε > 0 small, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M78">View MathML</a>

Case 2. q = p.

In this case, we have that h'(t) = 0 if and only if,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M79">View MathML</a>

Since we suppose λ, δ ∈ (0,Λ1), we can use Poincaré's inequality to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M80">View MathML</a>

Thus, there exists tε > 0 satisfying (2.14).

Arguing as in the first case we conclude that, from (2.16) for ε > 0 small, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M81">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M82">View MathML</a> if N > p2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M83">View MathML</a> if N = p2, then εp-1 = o(εp-1| ln ε|). If N > p2, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M84">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M85">View MathML</a>. Choosing ε > 0 small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M86">View MathML</a>

This concludes the proof.

By Lemmas 2.1 and 2.2 we can prove our first result.

Proof of Theorem 1.1.

Since Iλ,δ satisfies the geometric conditions of the Mountain-Pass theorem, there exists {(um, vm)} ⊂ E such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M87">View MathML</a>. It follows from Lemmas 2.1 and 2.2 that {(um, vm)} converges, along a subsequence, to a nonzero critical point (u,v) ∈ E of Iλ,δ. Then, if we denote by u- = max{-u,0} and v- = max{-v,0} the negative part of u and v, respectively, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M88">View MathML</a>

it follows that (u-,v-) = (0,0). Hence, u,v ≥ 0 in Ω. The Theorem 1.1 is proved.

We finalize this section with the study of the asymptotic behavior of the minimax level cλ,δ as both the parameters λ, δ approach zero.

Lemma 2.3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M89">View MathML</a>.

Proof. We first prove the second equality. It follows from λ = δ = 0 that λ|u|q + δ|v|q ≡ 0. If A, B, uε, gε, and tε are the same as those in the proof of Lemma 2.2, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M90">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M91">View MathML</a>

Taking the limit as ε →0+ and using (2.11), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M251">View MathML</a>.

In order to obtain the reverse inequality we consider {(um, vm)} ⊂ E such that I0,0 (um, vm) → c0,0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M93">View MathML</a>. It is easy to show that the sequence {(um, vm)} is bounded in E and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M94">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M95">View MathML</a>

Taking the limit in the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M96">View MathML</a> we conclude, as in the proof of Lemma 2.1, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M97">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M98">View MathML</a>

and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M92">View MathML</a>.

We proceed now with the calculation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M99">View MathML</a>. Let {λm},{δm} ⊂ ℝ+ such that λm, δm → 0+. Since λm, δm are positive, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M100">View MathML</a> whenever (u, v) is nonnegative. Thus, for this kind of function, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M101">View MathML</a>.

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M102">View MathML</a>

in the last equality, we have used the infimum c0,0 which can be attained at a nonnegative solution. The above inequality implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M103">View MathML</a>

(2.17)

On the other hand, it follows from Theorem 1.1 that there exists {(um, vm)} ⊂ E such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M104">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M105">View MathML</a> is bounded, the same argument performed in the proof of Lemma 2.1 implies that {(um, vm)} is bounded in E. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M106">View MathML</a>

(2.18)

Let tm > 0 be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M107">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M108">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M109">View MathML</a>

If {tm} is bounded, we can use the above estimate and (2.18) to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M110">View MathML</a>

This and (2.17) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M111">View MathML</a>

that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M112">View MathML</a>.

It remains to check that {tm} is bounded. A straightforward calculation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M113">View MathML</a>

(2.19)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M108">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M114">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M115">View MathML</a>, and therefore from the above expression it follows that ∫Ω F(x, um, vm)dx C5 > 0. Thus, the boundedness of {(um, vm)} and (2.19) imply that {tm} is bounded. This completes the proof.

3 Some technical lemmas

In this section, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M116">View MathML</a> the Banach space of finite Radon measures over Ω equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M117">View MathML</a>. A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M118">View MathML</a> is said to converge weakly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M119">View MathML</a> provided σm(φ) → σ(φ) for all φ C0(Ω). By [18, Theorem 1.39], every bounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M118">View MathML</a> contains a weakly convergent subsequence.

The next lemma is a version of the second concentration-compactness lemma of Lions [21]. It is also inspired by [18, Lemma 1.40] and [[22], Lemma 2.4].

Lemma 3.1. Suppose that the sequence {(um,vm)} ⊂ D1,p(ℝN) × D1,p(ℝN) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M120">View MathML</a>

and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M121">View MathML</a>

(3.1)

then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M122">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M123">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M124">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M125">View MathML</a>

(3.5)

Moreover, if (u,v) = (0,0) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126">View MathML</a>, then the measures μ,ν, and σ are concentrated at a single point, respectively.

Proof. We first recall that, in view of the definition of SF, for each nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M127">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M128">View MathML</a>

Moreover, arguing as [8, Lemma 5], we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M129">View MathML</a>

Since F is p*-homogeneous, we can use the two above expressions and argue along the same line of the proof of Lemma 1.40 in [18] to conclude that (3.2)-(3.5) hold. If (u, v) = (0,0) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126">View MathML</a>, the same argument of step 3 of the proof of Lemma 1.40 in [18] implies that the measures μ, ν and σ are concentrated at a single point, respectively.

Remark 3.1. We notice that the last conclusion of the above result holds even if (u, v) ≢ (0,0). Indeed, in this case we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M43">View MathML</a>and notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M130">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M131">View MathML</a>and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M132">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M133">View MathML</a>, where μ,σ, and ν are the same as those in Lemma 3.1. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126">View MathML</a>we also have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M134">View MathML</a>and the result follows from the last part of Lemma 3.1.

Now, we introduce the following Lemma.

Lemma 3.2. Suppose {(um, vm)} ⊂ E such that Ω F(x, um, vm)dx = 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M135">View MathML</a>. Then there exist {rm} ⊂ (0, +∞) and {ym} ⊂ ℝN such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M136">View MathML</a>

(3.6)

contains a convergent subsequence denoted again by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M137">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M138">View MathML</a>in D1,p(ℝN) × D1,p(ℝN). Moreover, as m → ∞, we have rm → 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M139">View MathML</a>.

Proof. For each r > 0, we consider the Lévy concentration functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M140">View MathML</a>

Since for every m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M141">View MathML</a>

there exist rm > 0 and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M142">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M143">View MathML</a>

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M144">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M145">View MathML</a> is bounded. Hence, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M146">View MathML</a> and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M147">View MathML</a>

We shall prove that the above sequences {rm} and {ym} satisfy the statements of the lemma. First notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M148">View MathML</a>

(3.7)

By (3.6), a straightforward calculation provides

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M149">View MathML</a>

Hence, we can apply Lemma 3.1 to obtain (ω1,ω2) ∈ D1,p(ℝN) × D1,p (ℝN) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M150">View MathML</a>

(3.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M125">View MathML</a>

(3.9)

The second equality in (3.8) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M151">View MathML</a>. If one of these values belongs to the open interval (0,1), we can use (3.8), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M152">View MathML</a> and (3.9) to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M153">View MathML</a>

which is a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M154">View MathML</a>. Actually, it follows from (3.7) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M155">View MathML</a> for any R > 1. Thus, we conclude that ν= 0.

Let us prove that ||ν|| = 0. Arguing by contradiction, then ||ν|| = 1. It follows from the first equality in (3.8) that SF ≥ ||μ|| + ||σ||. On the other hand, the first inequality in (3.9) provides ||μ|| + ||σ|| ≥ SF. Hence, we conclude that ||μ|| + ||σ|| = SF. Since we suppose that ||ν|| = 1 we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M126">View MathML</a>. It follows from Remark 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M156">View MathML</a> for some x0 ∈ ℝN. Thus, from (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M157">View MathML</a>

This contradiction proves that ∥ν∥ = 0.

Since ∥ν∥ = ν= 0, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M158">View MathML</a>. This and (3.8) provide

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M159">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M160">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M161">View MathML</a> strongly in D1,p(ℝN) × D1,p(ℝN) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M162">View MathML</a> for a.e. x ∈ ℝN. In order to conclude the proof we notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M163">View MathML</a>

Since {(um, vm)} is bounded and (ω1, ω2) ≢ (0,0), we infer from the above equality that, up to a subsequence, rm r0 ≥ 0. If |ym| → ∞, for each fixed x ∈ ℝN, we have that there exists mx N such that rmx + ym ∉ Ω for m mx. For such values of m we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M164">View MathML</a>. Taking the limit and recalling that x ∈ ℝ is arbitrary, we conclude that (ω1, ω2) = (0,0), which is a contradiction. So, along a subsequence, ym y ∈ ℝN.

We claim that r0 = 0. Indeed, suppose by contradiction that r0 > 0. Then, as m becomes large, the set Ωm = (Ω-ym)/rm approaches Ω0 = (Ω -y)/r0 ≠ ℝN. This implies that ω1,ω2 has compact support in ℝN. On the other hand, since (ω1,ω2) achieves the infimum in (2.3) and F is homogeneous, we can use the Lagrange Multiplier Theorem to conclude that ( ω1, ω2) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M165">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M166">View MathML</a>. It follows from (F2) and the maximum principle that at least one of the functions ω1,ω2 is positive in ℝN. But this contradicts supp (ω1,ω2) ⊂ Ω0. Hence, we conclude that r0 = 0. Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M167">View MathML</a> we obtain rmx + ym ∉ Ω for large values of m, and therefore we should have (ω1, ω2) ≡ (0, 0) again. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M168">View MathML</a> and the proof is completed.

Up to translations, we may assume that 0 ∈ Ω, since Ω is a smooth bounded domain of ℝN, we can choose r > 0 small enough such that Br = Br(0) = {x ∈ ℝN : d(x, 0) < r} ⊂ Ω and the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M169">View MathML</a>

are homotopically equivalent to Ω. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M171">View MathML</a>

We define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M172">View MathML</a>

and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M173">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M174">View MathML</a>

Clearly, mλ,δ is nonincreasing in λ, δ. Note that mλ,δ > 0 for all λ, δ > 0.

Arguing as in the proof of Lemma 2.3 and Theorem 1.1, we obtain the following result.

Lemma 3.3. Suppose F satisfies (F0)-(F2), then the infimum mλ,δ is attained by a nonneg-ative radial function (uλ,δ, vλ,δ) ∈ Erad whenever 2 ≤ p < q < p* and λ,δ > 0, or q = p and λ,δ ∈ (0,Λ1,rad), where Λ1,rad > 0 is the first eigenvalue of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M175">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M176">View MathML</a>

We introduce the barycenter map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M177">View MathML</a> as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M178">View MathML</a>

This map has the following property.

Lemma 3.4. If N p2,2 ≤ p q < p* and F satisfies (F0)-(F2), then there exists λ* > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M179">View MathML</a>whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M180">View MathML</a>and Iλ,δ(u, v) ≤ mλ,δ.

Proof. By way of contradiction, we suppose that there exist {λm}, {δm} ⊂ ℝ+ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M181">View MathML</a> such that λm, δm → 0+ as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M182">View MathML</a> but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M183">View MathML</a>.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M184">View MathML</a> we have that {(um, vm)} is bounded in E. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M185">View MathML</a>

Since λm, δm → 0+, we can use the boundedness of {(um, vm)} to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M186">View MathML</a>

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M187">View MathML</a>

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M188">View MathML</a>

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M189">View MathML</a> both converge to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M190">View MathML</a>, we can use the above expression and ∫(λm|um|q + δm|vm|q)dx → 0 again to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M191">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M192">View MathML</a>

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M193">View MathML</a> and notice that tm(um, vm) satisfies the hypotheses of Lemma 3.2. Using Lemma 3.2, there exist sequences {rm} ⊂ (0,+∞) and {ym} ⊂ ℝN satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M194">View MathML</a> we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M195">View MathML</a> in D1,p (ℝN) × D1,p (ℝN).

The definition of β(u, v), (3.10), the strong convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M196">View MathML</a> and Lebesgue's theorem provide

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M197">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M198">View MathML</a> and ∫F(x,ω1,ω2)dx = 1, the above expression implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M199">View MathML</a>

which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M183">View MathML</a>.

According to Lemma 3.3, for each λ, δ > 0 small the infimum mλ,δ is attained by a nonnegative radial function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M200">View MathML</a>. We consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M201">View MathML</a>

and define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M202">View MathML</a> by setting, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M203">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M204">View MathML</a>

(3.11)

A change of variables and straightforward calculations show that the map γ is well defined. Since σλ,δ is radial, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M205">View MathML</a>. Hence, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M203">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M206">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M207">View MathML</a>.

Along the way of proving Lemma 3.4 we can check easily the following

Lemma 3.5. If λ,δ → 0+, αλ,δ → 1.

Proof. By Lemma 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M208">View MathML</a>

As before <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M209">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M210">View MathML</a>, the above expression and the same arguments used in the proof of Lemma 3.3 imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M211">View MathML</a>

The above equality and the definition of αλ,δ imply that αλ,δ → 1. The lemma is proved.

Next we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M212">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M213">View MathML</a>

We have the following

Lemma 3.6. if F satisfies (F0)-(F2), then there exists λ** > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M214">View MathML</a>

(3.12)

for all λ, δ ∈ (0, λ**).

Proof. Arguing by contradiction, we suppose that there exist sequences {λm},{δm} ⊂ ℝ+ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M215">View MathML</a> such that λm, δm → 0+, as m → ∞, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M216">View MathML</a> for all m. Up to a subsequence tm t0 ∈ [0, 1]. Moreover, the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M217">View MathML</a> and Lemma 3.4 imply that, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M218">View MathML</a>. From Lemma 3.5 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M219">View MathML</a>. So, we can use the definition of Hλ,δ to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M220">View MathML</a>, which is a contradiction. The lemma is proved.

4 Proof of Theorem 1.2

We begin with the following lemma.

Lemma 4.1. If (u, v) is a critical point of Iλ,δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M27">View MathML</a>, then it is a critical point of Iλ,δ in E.

Proof. The proof is almost the same as that [4, Lemma 4.1] and is omitted here.

Lemma 4.2. Suppose F satisfies (F0)-(F2), then any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M221">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M35">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34">View MathML</a>contains a convergent subsequence for λ,δ > 0 if q > p and λ,δ ∈ (0, λ*) if q = p for some small λ* > 0.

Proof. By hypothesis there exists a sequence θm ∈ ℝ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M222">View MathML</a> as m → ∞, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M223">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M224">View MathML</a>

Recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M225">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M226">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M227">View MathML</a>

Consequently ∥(um,vm)∥E → 0.

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M228">View MathML</a> it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M229">View MathML</a>

for some C > 0. Hence we arrive at a contradiction if λ, δ > 0 and q > p or λ, δ ∈ (0, λ*) for small λ* > 0 when q = p. Thus we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M230">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M231">View MathML</a>, we conclude that θm = 0, consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M34">View MathML</a>. Using this information we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M232">View MathML</a>

so by Lemma 2.1 the proof is completed.

Below we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M233">View MathML</a> the restriction of Iλ,δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M234">View MathML</a>.

Lemma 4.3. Suppose N p2,2 ≤ p q < p* and F satisfies (F0)-(F2), let Λ = min{λ*,λ**} > 0, λ, δ ∈ (0,Λ), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M235">View MathML</a>, where λ*, λ** given by Lemmas 34 and 3.6, respectively.

Proof. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M236">View MathML</a>, where Aj,j = 1,2,...,m, are closed and contractible sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M237">View MathML</a>, i.e., there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M238">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M239">View MathML</a>

where ϑ Aj is fixed. Consider Bj = γ-1(Aj), 1 ≤ j m. The sets Bj are closed and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M240">View MathML</a>

We define the deformation gj : [0, 1] × Bj by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M241">View MathML</a>

for λ,δ ∈ (0,Λ). Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M242">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M243">View MathML</a>

and gj(1,y) = Hλ,δ(1, hj(1,γ (y))) = β(hj(1,γ(y))) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M244">View MathML</a>

Thus the sets Bj are contractible in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M245">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M246">View MathML</a>.

Proof of Theorem 1.2.

Using Lemmas 2.1, 2.2, and 3.3 we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M247">View MathML</a> for λ,δ ∈ (0,Λ). Moreover, by Lemma 4.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M248">View MathML</a> satisfies the (PS)c condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M249">View MathML</a>. Therefore, by Lemma 4.3, a standard deformation argument implies that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M250">View MathML</a> contains at least catΩ(Ω) critical points of the restriction of Iλ,δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/27/mathml/M234">View MathML</a>. Now Lemma 4.1 implies that Iλ,δ has at least catΩ(Ω) critical points, and therefore at least catΩ(Ω) nontrivial solutions of (1.1). As Theorem 1.1, the obtained solutions are nonnegative in Ω. The proof is completed.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This study was supported by the Youth Foundation of Hubei Engineering University (No. Z2012003).

References

  1. Brezis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math. 36, 437–477 (1983). Publisher Full Text OpenURL

  2. Rey, O: A multiplicity results for a variational problem with lack of compactness. Nonlinear Anal. 13(10), 1241–1249 (1989). Publisher Full Text OpenURL

  3. Alves, CO, Ding, YH: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J Math Anal Appl. 279(2), 508–521 (2003). Publisher Full Text OpenURL

  4. Ding, L, Xiao, SW: Multiple positive solutions for a critical quasilinear elliptic system. Nonlinear Anal. 72, 2592–2607 (2010). Publisher Full Text OpenURL

  5. Hsu, TS: Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities. Nonlinear Anal. 71(7-8), 2688–2698 (2009). Publisher Full Text OpenURL

  6. Shen, Y, Zhang, JH: Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent. Nonlinear Anal. 74, 1019–1030 (2011). Publisher Full Text OpenURL

  7. Han, PG: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houston J Math. 32, 1241–1257 (2006)

  8. Filho, DCM, Souto, MAS: Systems of p-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Comm Partial Diff Equ. 24, 1537–1553 (1999). Publisher Full Text OpenURL

  9. Barbosa, ER, Montenegro, M: Nontrivial solutions for critical potential elliptic systems. J Diff Equ. 250, 3398–3417 (2011). Publisher Full Text OpenURL

  10. Alves, CO, Filho, DCM, Miyagaki, OH: Multiple solutions for an elliptic system on bounded or unbounded domains. Nonlinear Anal. 56, 555–568 (2004). Publisher Full Text OpenURL

  11. Kang, DS, Peng, SJ: Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents. Sci China Math. 54, 243–256 (2011). Publisher Full Text OpenURL

  12. Figueiredo, GM, Furtado, MF: Multiple positive solutions for a quasilinear system of Schröodinger equations. NoDEA Nonlinear Diff Equ Appl. 15, 309–333 (2008). Publisher Full Text OpenURL

  13. Lü, DF: Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent. Nonlinear Anal. 74, 6371–6382 (2011). Publisher Full Text OpenURL

  14. Chu, CM, Tang, CL: Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents. Nonlinear Anal. 71, 5118–5130 (2009). Publisher Full Text OpenURL

  15. Adriouch, K, El Hamidi, A: The Nehari manifold for systems of nonlinear elliptic equations. Nonlinear Anal. 64, 2149–2167 (2006). Publisher Full Text OpenURL

  16. Lü, DF, Xiao, JH: Multiple solutions for weighted nonlinear elliptic system involving critical exponents. Math Comput Model. 55, 816–827 (2012). Publisher Full Text OpenURL

  17. Ribeiro, B: The Ambrosetti-Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity. J Math Anal Appl. 363, 606–617 (2010). Publisher Full Text OpenURL

  18. Willem, M: Minimax Theorem. Birkhäuser, Boston (1996)

  19. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc Am Math Soc. 88, 486–490 (1983)

  20. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Am Math Soc. 352(12), 5703–5743 (2000). Publisher Full Text OpenURL

  21. Lions, PL: The concentration compactness principle in the calculus of variations. The limit case I Rev Mat Iberoam. 1, 145–201 (1985)

  22. Furtado, MF: Multiplicity of nodal solutions for a critical quasilinear equation with symmetry. Nonlinear Anal. 63, 1153–1166 (2005). Publisher Full Text OpenURL