Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems

Boundary Value Problems20102010:139126

DOI: 10.1155/2010/139126

Received: 23 November 2009

Accepted: 15 June 2010

Published: 6 July 2010

Abstract

Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq1_HTML.gif -Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq2_HTML.gif ik spectrum of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq3_HTML.gif -Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.

1. Introduction

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq4_HTML.gif be a bounded domain with Lipschitz boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq5_HTML.gif . We consider the following nonlinear elliptic boundary value problem. Find http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq6_HTML.gif and constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq7_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq8_HTML.gif is the negative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq9_HTML.gif -Laplacian, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq10_HTML.gif denotes the outer normal derivative of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq11_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq12_HTML.gif as well as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq13_HTML.gif are the positive and negative parts of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq14_HTML.gif , respectively. The nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq16_HTML.gif are some Carathéodory functions which are bounded on bounded sets. For reasons of simplification, we drop the notation for the trace operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq17_HTML.gif which is used on the functions defined on the boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq18_HTML.gif .

The motivation of our study is a recent paper of the author in [1] in which problem (1.1) was treated in case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq19_HTML.gif . We extend this approach and prove the existence of multiple solutions for the more general problem (1.1). To be precise, the existence of a smallest positive solution, a greatest negative solution, as well as a sign-changing solution of problem (1.1) is proved by using variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma. Additionally, the Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq20_HTML.gif ik spectrum for the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq21_HTML.gif -Laplacian takes an important part in our treatments.

Neumann boundary value problems in the form of (1.1) arise in different areas of pure and applied mathematics, for example, in the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [2, 3]), in the study of optimal constants for the Sobolev trace embedding (see [47]), or at non-Newtonian fluids, flow through porus media, nonlinear elasticity, reaction diffusion problems, glaciology, and so on (see [811]).

The existence of multiple solutions for Neumann problems like those in the form of (1.1) has been studied by a number of authors, such as, for example, the authors of [1215], and homogeneous Neumann boundary value problems were considered in [16, 17] and [15], respectively. Analogous results for the Dirichlet problem have been recently obtained in [1821]. Further references can also be found in the bibliography of [1].

In our consideration, the nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq23_HTML.gif only need to be Carathéodory functions which are bounded on bounded sets whereby their growth does not need to be necessarily polynomial. The novelty of our paper is the fact that we do not need differentiability, polynomial growth, or some integral conditions on the mappings http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq25_HTML.gif .

First, we have to make an analysis of the associated spectrum of (1.1). The Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq26_HTML.gif ik spectrum for the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq27_HTML.gif -Laplacian with a nonlinear boundary condition is defined as the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq28_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq29_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ2_HTML.gif
(1.2)
has a nontrivial solution. In view of the identity
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ3_HTML.gif
(1.3)
we see at once that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq30_HTML.gif problem (1.2) reduces to the Steklov eigenvalue problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ4_HTML.gif
(1.4)
We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq31_HTML.gif is an eigenvalue if (1.4) has nontrivial solutions. The first eigenvalue http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq32_HTML.gif is isolated and simple and has a first eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq33_HTML.gif which is strictly positive in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq34_HTML.gif (see [22]). Furthermore, one can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq35_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq36_HTML.gif (cf., [23, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq37_HTML.gif and Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq38_HTML.gif ] or [24, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq39_HTML.gif ]), and along with the results of Lieberman in [25, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq40_HTML.gif ] it holds that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq41_HTML.gif . This fact combined with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq42_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq43_HTML.gif yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq44_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq45_HTML.gif denotes the interior of the positive cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq46_HTML.gif in the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq47_HTML.gif , given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ5_HTML.gif
(1.5)
Let us recall some properties of the Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq48_HTML.gif ik spectrum. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq49_HTML.gif is an eigenvalue of (1.4), then the point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq50_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq51_HTML.gif . Since the first eigenfunction of (1.4) is positive, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq52_HTML.gif clearly contains the two lines http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq54_HTML.gif . A first nontrivial curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq55_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq56_HTML.gif through http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq57_HTML.gif was constructed and variationally characterized by a mountain-pass procedure by Martínez and Rossi [26]. This yields the existence of a continuous path in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq58_HTML.gif joining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq60_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq61_HTML.gif is above the curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq62_HTML.gif . The functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq63_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq64_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ6_HTML.gif
(1.6)
Due to the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq65_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq66_HTML.gif , there exists a variational characterization of the second eigenvalue of (1.4) meaning that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq67_HTML.gif can be represented as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ7_HTML.gif
(1.7)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ8_HTML.gif
(1.8)

The proof of this result is given in [26].

An important part in our considerations takes the following Neumann boundary value problem defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ9_HTML.gif
(1.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq68_HTML.gif is a constant. As pointed out in [1], there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq69_HTML.gif of problem (1.9) which is required for the construction of sub- and supersolutions of problem (1.1).

2. Notations and Hypotheses

Now, we impose the following conditions on the nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq71_HTML.gif in problem (1.1). The maps http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq73_HTML.gif are Carathéodory functions, which means that they are measurable in the first argument and continuous in the second one. Furthermore, we suppose the following assumptions.

(H) (f1)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ10_HTML.gif
(2.1)
(f2)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ11_HTML.gif
(2.2)

(f3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq77_HTML.gif is bounded on bounded sets.

(f4) There exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq79_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ12_HTML.gif
(2.3)
(H) (g1)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ13_HTML.gif
(2.4)
(g2)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ14_HTML.gif
(2.5)

(g3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq83_HTML.gif is bounded on bounded sets.

(g4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq85_HTML.gif satisfies the condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ15_HTML.gif
(2.6)

for all pairs http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq86_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq87_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq88_HTML.gif is a positive constant and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq89_HTML.gif .

(H) Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq91_HTML.gif be above the first nontrivial curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq92_HTML.gif of the Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq93_HTML.gif ik spectrum constructed in [26] (see Figure 1).

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Fig1_HTML.jpg
Figure 1

Fu http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq94_HTML.gif ik spectrum

Note that (H2)(g4) implies that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq95_HTML.gif fulfills a condition as in (H2)(g4), too. Moreover, we see at once that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq96_HTML.gif is a trivial solution of problem (1.1) because of the conditions (H1)(f1) and (H2)(g1), which guarantees that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq97_HTML.gif . It should be noted that hypothesis (H3) includes that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq98_HTML.gif (see [26] or Figure 1).

Example 2.1.

Let the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq100_HTML.gif be given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ16_HTML.gif
(2.7)

Then all conditions in (H1)(f1)–(f4) and (H2)(g1)–(g4) are fulfilled.

Definition 2.2.

A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq101_HTML.gif is called a weak solution of (1.1) if the following holds:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ17_HTML.gif
(2.8)

Definition 2.3.

A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq102_HTML.gif is called a subsolution of (1.1) if the following holds:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ18_HTML.gif
(2.9)

Definition 2.4.

A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq103_HTML.gif is called a supersolution of (1.1) if the following holds:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ19_HTML.gif
(2.10)

We recall that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq104_HTML.gif denotes all nonnegative functions of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq105_HTML.gif . Furthermore, for functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq106_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq107_HTML.gif , we have the relation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq108_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq109_HTML.gif stands for the well-known trace operator.

3. Extremal Constant-Sign Solutions

For the rest of the paper we denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq110_HTML.gif the first eigenfunction of the Steklov eigenvalue problem (1.4) corresponding to its first eigenvalue http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq111_HTML.gif . Furthermore, the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq112_HTML.gif stands for the unique solution of the auxiliary Neumann boundary value problem defined in (1.9). Our first lemma reads as follows.

Lemma 3.1.

Let conditions (H1)-(H2) be satisfied and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq113_HTML.gif . Then there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq116_HTML.gif are a positive supersolution and a negative subsolution, respectively, of problem (1.1).

Proof.

Setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq117_HTML.gif with a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq118_HTML.gif to be specified and considering the auxiliary problem (1.9), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ20_HTML.gif
(3.1)
In order to satisfy Definition 2.4 for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq119_HTML.gif , we have to show that the following inequality holds true meaning:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ21_HTML.gif
(3.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq120_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq121_HTML.gif . Condition (H1)(f2) implies the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq122_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ22_HTML.gif
(3.3)
and due to (H1)(f3), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ23_HTML.gif
(3.4)
Hence, we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ24_HTML.gif
(3.5)
Because of hypothesis (H2)(g2), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq123_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ25_HTML.gif
(3.6)
and thanks to condition (H2)(g3), we find a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq124_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ26_HTML.gif
(3.7)
Finally, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ27_HTML.gif
(3.8)
Using the inequality in (3.5) to the first integral in (3.2) yields
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ28_HTML.gif
(3.9)
which proves its nonnegativity if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq125_HTML.gif . Applying (3.8) to the second integral in (3.2) ensures that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ29_HTML.gif
(3.10)
We take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq126_HTML.gif to verify that both integrals in (3.2) are nonnegative. Hence, the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq127_HTML.gif is in fact a positive supersolution of problem (1.1). In a similar way one proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq128_HTML.gif is a negative subsolution, where we apply the following estimates:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ30_HTML.gif
(3.11)

This completes the proof.

The next two lemmas show that constant multipliers of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq129_HTML.gif may be sub- and supersolution of (1.1). More precisely, we have the following result.

Lemma 3.2.

Assume that (H1)-(H2) are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq130_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq131_HTML.gif sufficiently small and any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq132_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq133_HTML.gif is a positive subsolution of problem (1.1).

Proof.

The Steklov eigenvalue problem (1.4) implies for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq134_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ31_HTML.gif
(3.12)
Definition 2.3 is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq135_HTML.gif provided that the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ32_HTML.gif
(3.13)
is valid for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq136_HTML.gif . With regard to hypothesis (H1)(f4), we obtain, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq137_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ33_HTML.gif
(3.14)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq138_HTML.gif denotes the usual supremum norm. Thanks to condition (H2)(g1), there exists a number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq139_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ34_HTML.gif
(3.15)
In case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq140_HTML.gif we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ35_HTML.gif
(3.16)

Selecting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq141_HTML.gif guarantees that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq142_HTML.gif is a positive subsolution.

The following lemma on the existence of a negative supersolution can be proved in a similar way.

Lemma 3.3.

Assume that (H1)-(H2) are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq143_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq144_HTML.gif sufficiently small and any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq145_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq146_HTML.gif is a negative supersolution of problem (1.1).

Concerning Lemmas 3.1–3.3, we obtain a positive pair http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq147_HTML.gif and a negative pair http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq148_HTML.gif of sub- and supersolutions of problem (1.1) provided that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq149_HTML.gif is sufficiently small.

In the next step we are going to prove the regularity of solutions of problem (1.1) belonging to the order intervals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq151_HTML.gif , respectively. We also point out that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq152_HTML.gif is both a subsolution and a supersolution because of the hypotheses (H1)(f1) and (H2)(g1).

Lemma 3.4.

Assume (H1)-(H2) and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq153_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq154_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq155_HTML.gif ) is a solution of problem (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq156_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq157_HTML.gif , then it holds that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq158_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq159_HTML.gif ).

Proof.

We just show the first case; the other case acts in the same way. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq160_HTML.gif be a solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq161_HTML.gif . We directly obtain the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq162_HTML.gif -boundedness, and, hence, the regularity results of Lieberman in [25, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq163_HTML.gif ] imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq164_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq165_HTML.gif . Due to assumptions (H1)(f1), (H1)(f3), (H2)(g1), and (H2)(g3), we obtain the existence of constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq166_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ36_HTML.gif
(3.17)
Applying (3.17) to (1.1) provides
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ37_HTML.gif
(3.18)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq167_HTML.gif is a positive constant. We set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq168_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq169_HTML.gif and use Vázquez's strong maximum principle (cf., [27]) which is possible because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq170_HTML.gif . Hence, it holds that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq171_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq172_HTML.gif . Finally, we suppose the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq173_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq174_HTML.gif . Applying again the maximum principle yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq175_HTML.gif . However, because of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq176_HTML.gif in combination with the Neumann condition in (1.1), we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq177_HTML.gif . This is a contradiction and, hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq178_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq179_HTML.gif , which proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq180_HTML.gif .

The main result in this section about the existence of extremal constant-sign solutions is given in the following theorem.

Theorem 3.5.

Assume (H1)-(H2). For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq181_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq182_HTML.gif , there exists a smallest positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq183_HTML.gif of (1.1) in the order interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq184_HTML.gif with the constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq185_HTML.gif as in Lemma 3.1. For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq187_HTML.gif there exists a greatest solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq188_HTML.gif in the order interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq189_HTML.gif with the constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq190_HTML.gif as in Lemma 3.1.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq191_HTML.gif . Lemmas 3.1 and 3.2 guarantee that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq192_HTML.gif is a subsolution of problem (1.1) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq193_HTML.gif is a supersolution of problem (1.1). Moreover, we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq194_HTML.gif sufficiently small such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq195_HTML.gif . Applying the method of sub- and supersolution (see [28]) corresponding to the order interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq196_HTML.gif provides the existence of a smallest positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq197_HTML.gif of problem (1.1) fulfilling http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq198_HTML.gif . In view of Lemma 3.4, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq199_HTML.gif . Hence, for every positive integer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq200_HTML.gif sufficiently large, there exists a smallest solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq201_HTML.gif of problem (1.1) in the order interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq202_HTML.gif . We obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ38_HTML.gif
(3.19)

with some function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq203_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq204_HTML.gif .

Claim 1.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq205_HTML.gif is a solution of problem (1.1).

As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq207_HTML.gif , we obtain the boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq208_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq209_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq210_HTML.gif , respectively. Definition 2.2 holds, in particular, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq211_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq212_HTML.gif which results in
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ39_HTML.gif
(3.20)
with some positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq213_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq214_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq215_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq216_HTML.gif and due to the reflexivity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq217_HTML.gif we obtain the existence of a weakly convergent subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq218_HTML.gif . Because of the compact embedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq219_HTML.gif , the monotony of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq220_HTML.gif , and the compactness of the trace operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq221_HTML.gif , we get for the entire sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq222_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ40_HTML.gif
(3.21)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq223_HTML.gif solves problem (1.1), one obtains, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq224_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ41_HTML.gif
(3.22)
Setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq225_HTML.gif in (3.22) results in
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ42_HTML.gif
(3.23)
Using (3.21) and the hypotheses (H1)(f3) as well as (H2)(g3) yields
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ43_HTML.gif
(3.24)
which provides, by the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq226_HTML.gif -property of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq227_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq228_HTML.gif along with (3.21),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ44_HTML.gif
(3.25)

The uniform boundedness of the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq229_HTML.gif in conjunction with the strong convergence in (3.25) and conditions (H1)(f3) as well as (H2)(g3) admit us to pass to the limit in (3.22). This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq230_HTML.gif is a solution of problem (1.1).

Claim 2.

One has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq231_HTML.gif .

In order to apply Lemma 3.4, we have to prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq232_HTML.gif . Let us assume that this assertion is not valid meaning that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq233_HTML.gif . From (3.19) it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ45_HTML.gif
(3.26)
We set
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ46_HTML.gif
(3.27)
It is clear that the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq234_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq235_HTML.gif which ensures the existence of a weakly convergent subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq236_HTML.gif , denoted again by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq237_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ47_HTML.gif
(3.28)
with some function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq238_HTML.gif belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq239_HTML.gif . In addition, we may suppose that there are functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq240_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ48_HTML.gif
(3.29)
With the aid of (3.22), we obtain for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq241_HTML.gif the following variational equation:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ49_HTML.gif
(3.30)
We select http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq242_HTML.gif in the last equality to get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ50_HTML.gif
(3.31)
Making use of (3.17) in combination with (3.29) results in
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ51_HTML.gif
(3.32)
and, respectively,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ52_HTML.gif
(3.33)
We see at once that the right-hand sides of (3.32) and (3.33) belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq244_HTML.gif , respectively, which allows us to apply Lebesgue's dominated convergence theorem. This fact and the convergence properties in (3.28) show that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ53_HTML.gif
(3.34)
From (3.28), (3.31), and (3.34) we infer that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ54_HTML.gif
(3.35)
and the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq245_HTML.gif -property of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq246_HTML.gif corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq247_HTML.gif implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ55_HTML.gif
(3.36)
Remark that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq248_HTML.gif , which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq249_HTML.gif . Applying (3.26) and (3.36) along with conditions (H1)(f1), (H2)(g1) to (3.30) provides
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ56_HTML.gif
(3.37)

The equation above is the weak formulation of the Steklov eigenvalue problem in (1.4) where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq250_HTML.gif is the eigenfunction with respect to the eigenvalue http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq251_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq252_HTML.gif is nonnegative in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq253_HTML.gif , we get a contradiction to the results of Martínez and Rossi in [22, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq254_HTML.gif ] because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq255_HTML.gif must change sign on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq256_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq257_HTML.gif . Applying Lemma 3.4 yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq258_HTML.gif .

Claim 3.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq259_HTML.gif is the smallest positive solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq260_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq261_HTML.gif be a positive solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq262_HTML.gif . Lemma 3.4 immediately implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq263_HTML.gif . Then there exists an integer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq264_HTML.gif sufficiently large such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq265_HTML.gif . However, we already know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq266_HTML.gif is the smallest solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq267_HTML.gif which yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq268_HTML.gif . Passing to the limit proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq269_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq270_HTML.gif must be the smallest positive solution of (1.1). The existence of the greatest negative solution of (1.1) within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq271_HTML.gif can be proved similarly. This completes the proof of the theorem.

4. Variational Characterization of Extremal Solutions

Theorem 3.5 ensures the existence of extremal positive and negative solutions of (1.1) for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq272_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq273_HTML.gif denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq274_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq275_HTML.gif , respectively. Now, we introduce truncation functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq277_HTML.gif as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ57_HTML.gif
(4.1)
For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq278_HTML.gif the truncation operators on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq279_HTML.gif apply to the corresponding traces http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq280_HTML.gif . We just write for simplification http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq281_HTML.gif without http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq282_HTML.gif . Furthermore, the truncation operators are continuous, uniformly bounded, and Lipschitz continuous with respect to the second argument. By means of these truncations, we define the following associated functionals given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ58_HTML.gif
(4.2)

which are well defined and belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq283_HTML.gif . Due to the truncations, one can easily show that these functionals are coercive and weakly lower semicontinuous, which implies that their global minimizers exist. Moreover, they also satisfy the Palais-Smale condition.

Lemma 4.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq284_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq285_HTML.gif be the extremal constant-sign solutions of (1.1). Then the following hold.

(i) A critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq286_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq287_HTML.gif is a nonnegative solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq288_HTML.gif .

(ii) A critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq289_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq290_HTML.gif is a nonpositive solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq291_HTML.gif .

(iii) A critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq292_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq293_HTML.gif is a solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq294_HTML.gif .

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq295_HTML.gif be a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq296_HTML.gif meaning http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq297_HTML.gif . We have for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq298_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ59_HTML.gif
(4.3)
As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq299_HTML.gif is a positive solution of (1.1), it satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ60_HTML.gif
(4.4)
Subtracting (4.4) from (4.3) and setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq300_HTML.gif provide
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ61_HTML.gif
(4.5)

Based on the definition of the truncation operators, we see that the right-hand side of the equality above is equal to zero. On the other hand, the integrals on the left-hand side are strictly positive in case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq301_HTML.gif which is a contradiction. Thus, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq302_HTML.gif and, hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq303_HTML.gif . The proof for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq304_HTML.gif acts in a similar way which shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq305_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq306_HTML.gif , and therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq307_HTML.gif is a solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq308_HTML.gif . The statements in (i) and (ii) can be shown in the same way.

An important tool in our considerations is the relation between local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq309_HTML.gif -minimizers and local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq310_HTML.gif -minimizers for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq311_HTML.gif -functionals. The fact is that every local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq312_HTML.gif -minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq313_HTML.gif is a local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq314_HTML.gif -minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq315_HTML.gif which was proved in similar form in [1, Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq316_HTML.gif ]. This result reads as follows.

Proposition 4.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq317_HTML.gif is a local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq318_HTML.gif -minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq319_HTML.gif meaning that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq320_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ62_HTML.gif
(4.6)
then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq321_HTML.gif is a local minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq322_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq323_HTML.gif meaning that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq324_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ63_HTML.gif
(4.7)

We also refer to a recent paper (see [29]) in which the proposition above was extended to the more general case of nonsmooth functionals. With the aid of Proposition 4.2, we can formulate the next lemma about the existence of local and global minimizers with respect to the functionals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq325_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq326_HTML.gif .

Lemma 4.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq327_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq328_HTML.gif . Then the extremal positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq329_HTML.gif of (1.1) is the unique global minimizer of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq330_HTML.gif , and the extremal negative solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq331_HTML.gif of (1.1) is the unique global minimizer of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq332_HTML.gif . In addition, both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq333_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq334_HTML.gif are local minimizers of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq335_HTML.gif .

Proof.

As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq336_HTML.gif is coercive and weakly sequentially lower semicontinuous, its global minimizer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq337_HTML.gif exists meaning that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq338_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq339_HTML.gif . Concerning Lemma 4.1, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq340_HTML.gif is a nonnegative solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq341_HTML.gif . Due to condition (H2)(g1), there exists a number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq342_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ64_HTML.gif
(4.8)
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq343_HTML.gif and applying assumption (H1)(f4), inequality (4.8) along with the Steklov eigenvalue problem in (1.4) implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ65_HTML.gif
(4.9)

From the calculations above, we see at once that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq344_HTML.gif , which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq345_HTML.gif . This allows us to apply Lemma 3.4 getting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq346_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq347_HTML.gif is the smallest positive solution of (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq348_HTML.gif fulfilling http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq349_HTML.gif , it must hold that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq350_HTML.gif , which proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq351_HTML.gif is the unique global minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq352_HTML.gif . The same considerations show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq353_HTML.gif is the unique global minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq354_HTML.gif . In order to complete the proof, we are going to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq355_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq356_HTML.gif are local minimizers of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq357_HTML.gif as well. The extremal positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq358_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq359_HTML.gif , which means that there is a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq360_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq361_HTML.gif in the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq362_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq363_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq364_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq365_HTML.gif proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq366_HTML.gif is a local minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq367_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq368_HTML.gif . Applying Proposition 4.2 yields that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq369_HTML.gif is also a local http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq370_HTML.gif -minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq371_HTML.gif . Similarly, we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq372_HTML.gif is a local minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq373_HTML.gif , which completes the proof.

Lemma 4.4.

The functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq374_HTML.gif has a global minimizer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq375_HTML.gif which is a nontrivial solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq376_HTML.gif .

Proof.

As we know, the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq377_HTML.gif is coercive and weakly sequentially lower semicontinuous. Hence, it has a global minimizer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq378_HTML.gif . More precisely, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq379_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq380_HTML.gif which is a solution of (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq381_HTML.gif (see Lemma 4.1). The fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq382_HTML.gif (see the proof of Lemma 4.3) proves that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq383_HTML.gif is nontrivial meaning that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq384_HTML.gif .

5. Existence of Sign-Changing Solutions

The main result in this section about the existence of a nontrivial solution of problem (1.1) reads as follows.

Theorem 5.1.

Under hypotheses (H1)–(H3), problem (1.1) has a nontrivial sign-changing solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq385_HTML.gif .

Proof.

In view of Lemma 4.4, the existence of a global minimizer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq386_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq387_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq388_HTML.gif has been proved. This means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq389_HTML.gif is a nontrivial solution of (1.1) belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq390_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq391_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq392_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq393_HTML.gif must be a sign-changing solution because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq394_HTML.gif is the greatest negative solution and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq395_HTML.gif is the smallest positive solution of (1.1), which proves the theorem in this case. We still have to show the theorem in case that either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq396_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq397_HTML.gif . Let us only consider the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq398_HTML.gif because the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq399_HTML.gif can be proved similarly. The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq400_HTML.gif is a local minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq401_HTML.gif . Without loss of generality, we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq402_HTML.gif is a strict local minimizer; otherwise, we would obtain infinitely many critical points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq403_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq404_HTML.gif which are sign-changing solutions due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq405_HTML.gif and the extremality of the solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq406_HTML.gif . Under these assumptions, there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq407_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ66_HTML.gif
(5.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq408_HTML.gif . Now, we may apply the Mountain-Pass Theorem to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq409_HTML.gif (cf., [30]) thanks to (5.1) along with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq410_HTML.gif satisfies the Palais-Smale condition. This yields the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq411_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq412_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ67_HTML.gif
(5.2)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ68_HTML.gif
(5.3)
It is clear that (5.1) and (5.2) imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq413_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq414_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq415_HTML.gif is a sign-changing solution provided that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq416_HTML.gif . We have to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq417_HTML.gif which is fulfilled if there exists a path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq418_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ69_HTML.gif
(5.4)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq419_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq420_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq421_HTML.gif be equipped with the topologies induced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq422_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq423_HTML.gif , respectively. Furthermore, we set
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ70_HTML.gif
(5.5)
Because of the results of Martínez and Rossi in [26], there exists a continuous path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq424_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq425_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq426_HTML.gif is above the curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq427_HTML.gif of hypothesis (H3). Recall that the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq428_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ71_HTML.gif
(5.6)
This implies the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq429_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ72_HTML.gif
(5.7)
It is well known that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq430_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq431_HTML.gif , which implies the density of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq432_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq433_HTML.gif . Thus, a continuous path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq434_HTML.gif exists such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ73_HTML.gif
(5.8)
The boundedness of the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq435_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq436_HTML.gif ensures the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq437_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ74_HTML.gif
(5.9)
Theorem 3.5 yields that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq438_HTML.gif . Thus, for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq439_HTML.gif and any bounded neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq440_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq441_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq442_HTML.gif there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq443_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq444_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ75_HTML.gif
(5.10)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq445_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq446_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq447_HTML.gif . Using (5.10) along with a compactness argument implies the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq448_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ76_HTML.gif
(5.11)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq449_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq450_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq451_HTML.gif . Representing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq452_HTML.gif in terms of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq453_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ77_HTML.gif
(5.12)
In view of (5.11) we get for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq454_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq455_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ78_HTML.gif
(5.13)
Due to hypotheses (H1)(f1) and (H2)(g1), there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq456_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ79_HTML.gif
(5.14)
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq457_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq458_HTML.gif and using (5.14) provide
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ80_HTML.gif
(5.15)
Applying (5.15) to (5.13) yields
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ81_HTML.gif
(5.16)
We have constructed a continuous path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq459_HTML.gif joining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq460_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq461_HTML.gif . In order to construct continuous paths http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq462_HTML.gif connecting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq463_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq464_HTML.gif , respectively, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq465_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq466_HTML.gif , we first denote that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ82_HTML.gif
(5.17)

It holds that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq467_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq468_HTML.gif is a global minimizer of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq469_HTML.gif . By Lemma 4.1 the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq470_HTML.gif has no critical values in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq471_HTML.gif . The coercivity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq472_HTML.gif along with its property to satisfy the Palais-Smale condition allows us to apply the Second Deformation Lemma (see, e.g., [31, page 366]) to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq473_HTML.gif . This ensures the existence of a continuous mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq474_HTML.gif satisfying the following properties:

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq475_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq476_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq477_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq478_HTML.gif ,

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq479_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq480_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq481_HTML.gif .

Next, we introduce the path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq482_HTML.gif given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq483_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq484_HTML.gif which is obviously continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq485_HTML.gif joining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq486_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq487_HTML.gif . Additionally, one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ83_HTML.gif
(5.18)
Similarly, the Second Deformation Lemma can be applied to the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq488_HTML.gif . We get a continuous path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq489_HTML.gif connecting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq490_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq491_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ84_HTML.gif
(5.19)

In the end, we combine the curves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq492_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq493_HTML.gif to obtain a continuous path http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq494_HTML.gif joining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq495_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq496_HTML.gif . Taking into account (5.16), (5.18), and (5.19), we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq497_HTML.gif . This yields the existence of a nontrivial sign-changing solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq498_HTML.gif of problem (1.1) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq499_HTML.gif which completes the proof.

Authors’ Affiliations

(1)
Institut für Mathematik, Technische Universität Berlin

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© Patrick Winkert. 2010

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