Open Access

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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq2_HTML.gif ik spectrum of the https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq6_HTML.gif and constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq7_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq8_HTML.gif is the negative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq9_HTML.gif -Laplacian, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq10_HTML.gif denotes the outer normal derivative of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq11_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq12_HTML.gif as well as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq14_HTML.gif , respectively. The nonlinearities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq15_HTML.gif and https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq20_HTML.gif ik spectrum for the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq22_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq24_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq26_HTML.gif ik spectrum for the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq28_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq29_HTML.gif such that
https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ4_HTML.gif
(1.4)
We say that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq33_HTML.gif which is strictly positive in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq35_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq36_HTML.gif (cf., [23, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq37_HTML.gif and Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq38_HTML.gif ] or [24, Theorem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq40_HTML.gif ] it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq41_HTML.gif . This fact combined with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq42_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq43_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq44_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq46_HTML.gif in the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq47_HTML.gif , given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq48_HTML.gif ik spectrum. If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq50_HTML.gif belongs to https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq52_HTML.gif clearly contains the two lines https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq54_HTML.gif . A first nontrivial curve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq55_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq56_HTML.gif through https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq58_HTML.gif joining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq60_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq61_HTML.gif is above the curve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq62_HTML.gif . The functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq63_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq64_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq65_HTML.gif belongs to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq67_HTML.gif can be represented as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ7_HTML.gif
(1.7)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ9_HTML.gif
(1.9)

where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq71_HTML.gif in problem (1.1). The maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq72_HTML.gif and https://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)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ10_HTML.gif
(2.1)
(f2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ11_HTML.gif
(2.2)

(f3) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq79_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ12_HTML.gif
(2.3)
(H) (g1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ13_HTML.gif
(2.4)
(g2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ14_HTML.gif
(2.5)

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

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

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

(H) Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq91_HTML.gif be above the first nontrivial curve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq92_HTML.gif of the Fu https://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).

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

Fu https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq100_HTML.gif be given by
https://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 https://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:
https://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 https://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:
https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ19_HTML.gif
(2.10)

We recall that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq104_HTML.gif denotes all nonnegative functions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq105_HTML.gif . Furthermore, for functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq106_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq107_HTML.gif , we have the relation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq108_HTML.gif , where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq111_HTML.gif . Furthermore, the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq113_HTML.gif . Then there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq114_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq115_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq117_HTML.gif with a positive constant https://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
https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ21_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq120_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq122_HTML.gif such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ23_HTML.gif
(3.4)
Hence, we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq123_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq124_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ26_HTML.gif
(3.7)
Finally, we have
https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ29_HTML.gif
(3.10)
We take https://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 https://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 https://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:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq130_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq131_HTML.gif sufficiently small and any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq132_HTML.gif the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq134_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq135_HTML.gif provided that the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ32_HTML.gif
(3.13)
is valid for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq137_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ33_HTML.gif
(3.14)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq139_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ34_HTML.gif
(3.15)
In case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq140_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ35_HTML.gif
(3.16)

Selecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq141_HTML.gif guarantees that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq143_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq144_HTML.gif sufficiently small and any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq145_HTML.gif the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq147_HTML.gif and a negative pair https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq151_HTML.gif , respectively. We also point out that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq153_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq154_HTML.gif (resp., https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq156_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq157_HTML.gif , then it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq158_HTML.gif (resp., https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq161_HTML.gif . We directly obtain the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq163_HTML.gif ] imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq164_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq166_HTML.gif satisfying
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ37_HTML.gif
(3.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq167_HTML.gif is a positive constant. We set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq168_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq170_HTML.gif . Hence, it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq171_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq172_HTML.gif . Finally, we suppose the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq173_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq174_HTML.gif . Applying again the maximum principle yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq175_HTML.gif . However, because of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq177_HTML.gif . This is a contradiction and, hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq178_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq179_HTML.gif , which proves that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq182_HTML.gif , there exists a smallest positive solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq184_HTML.gif with the constant https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq186_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq187_HTML.gif there exists a greatest solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq188_HTML.gif in the order interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq189_HTML.gif with the constant https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq194_HTML.gif sufficiently small such that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq197_HTML.gif of problem (1.1) fulfilling https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq199_HTML.gif . Hence, for every positive integer https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq202_HTML.gif . We obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ38_HTML.gif
(3.19)

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

Claim 1.

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq207_HTML.gif , we obtain the boundedness of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq208_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq209_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq212_HTML.gif which results in
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ39_HTML.gif
(3.20)
with some positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq213_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq214_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq215_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq216_HTML.gif and due to the reflexivity of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq218_HTML.gif . Because of the compact embedding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq219_HTML.gif , the monotony of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq221_HTML.gif , we get for the entire sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq222_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ40_HTML.gif
(3.21)
Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq224_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ41_HTML.gif
(3.22)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq225_HTML.gif in (3.22) results in
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ43_HTML.gif
(3.24)
which provides, by the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq226_HTML.gif -property of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq227_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq228_HTML.gif along with (3.21),
https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq233_HTML.gif . From (3.19) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ45_HTML.gif
(3.26)
We set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq234_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq236_HTML.gif , denoted again by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq237_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ47_HTML.gif
(3.28)
with some function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq238_HTML.gif belonging to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq240_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq241_HTML.gif the following variational equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ49_HTML.gif
(3.30)
We select https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq242_HTML.gif in the last equality to get
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ51_HTML.gif
(3.32)
and, respectively,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq243_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ54_HTML.gif
(3.35)
and the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq245_HTML.gif -property of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq246_HTML.gif corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq247_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ55_HTML.gif
(3.36)
Remark that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq248_HTML.gif , which means that https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq251_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq252_HTML.gif is nonnegative in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq254_HTML.gif ] because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq255_HTML.gif must change sign on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq256_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq257_HTML.gif . Applying Lemma 3.4 yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq258_HTML.gif .

Claim 3.

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq260_HTML.gif .

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq262_HTML.gif . Lemma 3.4 immediately implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq263_HTML.gif . Then there exists an integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq264_HTML.gif sufficiently large such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq265_HTML.gif . However, we already know that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq267_HTML.gif which yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq268_HTML.gif . Passing to the limit proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq269_HTML.gif . Hence, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq273_HTML.gif denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq275_HTML.gif , respectively. Now, we introduce truncation functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq277_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ57_HTML.gif
(4.1)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq278_HTML.gif the truncation operators on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq279_HTML.gif apply to the corresponding traces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq280_HTML.gif . We just write for simplification https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq281_HTML.gif without https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq284_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq286_HTML.gif of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq288_HTML.gif .

(ii) A critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq289_HTML.gif of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq291_HTML.gif .

(iii) A critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq292_HTML.gif of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq294_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq295_HTML.gif be a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq296_HTML.gif meaning https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq297_HTML.gif . We have for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq298_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ59_HTML.gif
(4.3)
As https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq300_HTML.gif provide
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq302_HTML.gif and, hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq303_HTML.gif . The proof for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq305_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq306_HTML.gif , and therefore, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq309_HTML.gif -minimizers and local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq310_HTML.gif -minimizers for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq312_HTML.gif -minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq313_HTML.gif is a local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq314_HTML.gif -minimizer of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq317_HTML.gif is a local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq318_HTML.gif -minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq319_HTML.gif meaning that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq320_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ62_HTML.gif
(4.6)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq321_HTML.gif is a local minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq322_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq323_HTML.gif meaning that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq324_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq325_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq326_HTML.gif .

Lemma 4.3.

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

Proof.

As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq337_HTML.gif exists meaning that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq338_HTML.gif is a critical point of https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq342_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ64_HTML.gif
(4.8)
Choosing https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq344_HTML.gif , which means that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq346_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq348_HTML.gif fulfilling https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq349_HTML.gif , it must hold that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq350_HTML.gif , which proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq351_HTML.gif is the unique global minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq352_HTML.gif . The same considerations show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq353_HTML.gif is the unique global minimizer of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq355_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq356_HTML.gif are local minimizers of the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq357_HTML.gif as well. The extremal positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq358_HTML.gif belongs to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq360_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq361_HTML.gif in the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq362_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq363_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq364_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq365_HTML.gif proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq366_HTML.gif is a local minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq367_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq368_HTML.gif . Applying Proposition 4.2 yields that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq369_HTML.gif is also a local https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq370_HTML.gif -minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq371_HTML.gif . Similarly, we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq372_HTML.gif is a local minimizer of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq374_HTML.gif has a global minimizer https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq376_HTML.gif .

Proof.

As we know, the functional https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq378_HTML.gif . More precisely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq379_HTML.gif is a critical point of https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq383_HTML.gif is nontrivial meaning that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq386_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq387_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq388_HTML.gif has been proved. This means that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq390_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq391_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq392_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq394_HTML.gif is the greatest negative solution and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq396_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq397_HTML.gif . Let us only consider the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq398_HTML.gif because the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq399_HTML.gif can be proved similarly. The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq400_HTML.gif is a local minimizer of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq403_HTML.gif of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq405_HTML.gif and the extremality of the solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq406_HTML.gif . Under these assumptions, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq407_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ66_HTML.gif
(5.1)
where https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq411_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq412_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ67_HTML.gif
(5.2)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq413_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq414_HTML.gif . Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq416_HTML.gif . We have to show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq418_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ69_HTML.gif
(5.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq419_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq420_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq422_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq423_HTML.gif , respectively. Furthermore, we set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq424_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq425_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq426_HTML.gif is above the curve https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq428_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq429_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq430_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq431_HTML.gif , which implies the density of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq432_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq433_HTML.gif . Thus, a continuous path https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq434_HTML.gif exists such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq435_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq436_HTML.gif ensures the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq437_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq438_HTML.gif . Thus, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq439_HTML.gif and any bounded neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq440_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq441_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq442_HTML.gif there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq443_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq444_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ75_HTML.gif
(5.10)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq445_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq446_HTML.gif , and for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq448_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ76_HTML.gif
(5.11)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq449_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq450_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq451_HTML.gif . Representing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq452_HTML.gif in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq453_HTML.gif , we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq454_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq455_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq456_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ79_HTML.gif
(5.14)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq457_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq458_HTML.gif and using (5.14) provide
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq459_HTML.gif joining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq460_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq461_HTML.gif . In order to construct continuous paths https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq462_HTML.gif connecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq463_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq464_HTML.gif , respectively, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq465_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq466_HTML.gif , we first denote that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ82_HTML.gif
(5.17)

It holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq467_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq468_HTML.gif is a global minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq469_HTML.gif . By Lemma 4.1 the functional https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq471_HTML.gif . The coercivity of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq474_HTML.gif satisfying the following properties:

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

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

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

Next, we introduce the path https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq482_HTML.gif given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq483_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq484_HTML.gif which is obviously continuous in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq485_HTML.gif joining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq486_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq487_HTML.gif . Additionally, one has
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq488_HTML.gif . We get a continuous path https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq489_HTML.gif connecting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq490_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq491_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq492_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq493_HTML.gif to obtain a continuous path https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq494_HTML.gif joining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq495_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_IEq498_HTML.gif of problem (1.1) satisfying https://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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