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Boundary Value Problems

, 2009:540360 | Cite as

Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition

  • Fanglei Wang
  • Yukun An
Open Access
Research Article

Abstract

In this paper, we establish two different existence results of solutions for a nonlocal elliptic equations with nonlinear boundary condition. The first one is based on Galerkin method, and gives a priori estimate. The second one is based on Mountain Pass Lemma.

Keywords

Weak Solution Elliptic Equation Galerkin Method Dirichlet Boundary Condition Nontrivial Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we deal with the following elliptic equation with nonlinear boundary condition:

where Open image in new window is a bounded domain in Open image in new window with smooth boundary Open image in new window , Open image in new window , Open image in new window is the outer unite normal derivative, Open image in new window is continuous, Open image in new window , Open image in new window are Carathéodory functions.

For (1.1), if the nonlocal term Open image in new window is replaced by Open image in new window , then the equation
is related to the stationary analog of the Kirchhoff equation:

where Open image in new window . It was proposed by Kirchhoff [1] as an extension of the classical D'Alembert wave equations for free vibrations of elastic strings. The Kirchhoff model takes into account the length changes of the string produced by transverse vibrations. Equation (1.3) received much attention and an abstract framework to the problem was proposed after the work [2]. Some interesting and further results can be found in [3, 4] and the references therein. In addition, (1.2) has important physical and biological background. There are many authors who pay more attention to this equation. In particularly, authors concerned with the existence of solutions for (1.2) with zero Dirichlet boundary condition via Galerkin method, and built the variational frame in [5, 6]. More recently, Perera and Zhang obtained solutions of a class of nonlocal quasilinear elliptic boundary value problems using the variational methods, invariant sets of descent flow, Yang index, and critical groups [7, 8].

If the nonlocal term Open image in new window is replaced by Open image in new window , then the equation

arises in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitational (Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal runaway in Ohmic Heating, shear bands in metal deformed under high strain rates, among others. Because of its importance, in [9, 10], the authors similarly studied the existence of solution for (1.4) with zero Dirichlet boundary condition.

On the other hand, elliptic equations with nonlinear boundary conditions have become rather an active area of research; see [11, 12, 13, 14, 15] and reference therein. Those references present necessary and sufficient conditions of solutions of elliptic equations with nonlinear boundary conditions. In [13], the authors study the elliptic equation
with the nonlinear boundary condition

They obtain various existence results applying coincidence degree theory and the method of upper and lower solutions.

Inspired by the above references, we deal with the existence of solutions for elliptic equation (1.1) with nonlinear boundary condition based on Galerkin method and the Mountain Pass Lemma.

The paper is organized as follows. In Section 2, we will give the existence of solution for (1.1) via Galerkin method. In Section 3, we will study the solution for (1.1) using the Mountain Pass Lemma.

2. Existence

In this section, we state and prove the main theorem via Galerkin method when Open image in new window is bounded.

For convenience, we give the following hypotheses.

(H1)A typical assumption for Open image in new window is that there exists an Open image in new window such that Open image in new window , for all Open image in new window

where Open image in new window are constants, Open image in new window , Open image in new window .

(H3) The function Open image in new window is not identically zero.

Let Open image in new window be endowed with norm Open image in new window . Then Open image in new window is a Banach space.

A function Open image in new window is a weak solution of (1.1) if

for all Open image in new window .

Lemma 2.1.

Suppose that Open image in new window is a continuous function such that Open image in new window on Open image in new window , where Open image in new window is the usual inner product in Open image in new window and Open image in new window its related norm. Then, there exists Open image in new window such that Open image in new window .

Lemma 2.2 (see [16]).

Let Open image in new window be a domain in Open image in new window satisfying the uniform Open image in new window -regularity condition, and suppose that there exists a simple Open image in new window -extension operator Open image in new window for Open image in new window . Also suppose that Open image in new window and Open image in new window . Then

If Open image in new window , then the embedding still holds for Open image in new window . Moreover, if Open image in new window , then the embedding is compact.

Theorem 2.3.

Assume that (H1)–(H3) hold. In addition, we suppose that

Then problem (1.1) has at least one weak solution. Besides, any solution Open image in new window satisfies the estimate

Proof.

Let Open image in new window be different complete orthonormal systems for Open image in new window and set

Then Open image in new window is isometric to Open image in new window . Then, each Open image in new window is uniquely associated to Open image in new window by the relation Open image in new window . Since Open image in new window are, respectively, orthonormal in Open image in new window , we get Open image in new window .

We search for solutions Open image in new window of the approximate problem
To solve this algebraic system we define the operator Open image in new window

By condition (H2), the growth of function Open image in new window is subcritical, so Open image in new window defines a continuous Nemytskii mapping Open image in new window . Similarly, we also define a continuous mapping Open image in new window .

From the continuity of Open image in new window and Open image in new window , with respect to Open image in new window , we denote that Open image in new window is continuous. Therefore, from (H1), (H2), (H4) and Hölder's inequality, we note that Open image in new window
On the other hand, by Lemma 2.2, we have

where Open image in new window is constant.

From (2.9) and (2.10), we can prove that
This shows that there exists Open image in new window , depending only on Open image in new window , such that Open image in new window if Open image in new window . Then system (2.7) has a solution Open image in new window satisfying
From this bound estimate, going to a subsequence if necessary, there are Open image in new window and Open image in new window such that
Then fixing Open image in new window in (2.7) and letting Open image in new window , we conclude that
From the completeness of Open image in new window , identity holds with Open image in new window replaced by any Open image in new window . In particularly, when Open image in new window , we get
On the other hand, let Open image in new window in (2.7) and passing to the limit, we get
Then we conclude that Open image in new window , which shows that Open image in new window is a solution of (1.1). Finally, if Open image in new window is any solution of (1.1) and Open image in new window is nontrivial, then

The proof is complete.

3. Variational Method

In this section, we consider the following problem:

where Open image in new window are constants, and Open image in new window are defined in (H2).

The nontrivial solution of (3.1) comes from the Mountain Pass Lemma in [17].

Lemma 3.1 (Mountain Pass Lemma).

Let Open image in new window be a Banach space and let Open image in new window satisfy the Palais-Smale condition. Suppose also that

is a critical value of Open image in new window .

Theorem 3.2.

Assume the conditions (H1)–(H3) hold. In addition, the function Open image in new window satisfies

(H5)there exist Open image in new window with Open image in new window and Open image in new window , such that Open image in new window , Open image in new window , where Open image in new window .

Then (3.1) has a nontrivial solution.

Proof.

The weak solutions of (3.1) are critical points of the functional Open image in new window defined by

where Open image in new window .

Let us check the Open image in new window condition. Let Open image in new window , we have
Let Open image in new window be a Palais-Smale sequence in Open image in new window , that is, Open image in new window and Open image in new window and assume the contradiction that Open image in new window , then, from (H1), (H5), we have
where Open image in new window . Then by the Sobolev embedding theorem and Lemma 2.2, we can select Open image in new window such that

which is a contradiction with Open image in new window . Hence Open image in new window is bounded in Open image in new window . So Open image in new window admits a weakly convergence subsequence. From (H2), all the growth of Open image in new window is subcritical, so the standard argument shows that Open image in new window admits a strongly convergence subsequence.

Next we will verify the hypotheses of Lemma 3.1. By Hölder's inequality, Sobolev embedding theorem, and Lemma 2.2, we have
So we obtain

Let Open image in new window , then we take Open image in new window such that Open image in new window , when Open image in new window is sufficient small.

So for Open image in new window and Open image in new window small enough, then we have Open image in new window for all Open image in new window .

Since Open image in new window , we obtain Open image in new window when Open image in new window .

Let Open image in new window , with Open image in new window large enough, we have Open image in new window and Open image in new window . So by the Mountain Pass Lemma and (H3), we have a nontrivial solution Open image in new window for (3.1). The proof is complete.

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Copyright information

© F.Wang and Y. An. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingChina

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