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

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.

1. Introduction

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

(1.1)

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

For (1.1), if the nonlocal term is replaced by , then the equation

(1.2)

is related to the stationary analog of the Kirchhoff equation:

(1.3)

where . 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 is replaced by , then the equation

(1.4)

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 [11–15] 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

(1.5)

with the nonlinear boundary condition

(1.6)

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 is bounded.

For convenience, we give the following hypotheses.

(H1)A typical assumption for is that there exists an such that , for all

(H2) For all , assume that the functions , satisfying

(2.1)

where are constants, , .

(H3) The function is not identically zero.

Let be endowed with norm . Then is a Banach space.

A function is a weak solution of (1.1) if

(2.2)

for all .

Lemma 2.1.

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

Lemma 2.2 (see [16]).

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

(2.3)

If , then the embedding still holds for . Moreover, if , then the embedding is compact.

Theorem 2.3.

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

(H4)there exist constants such that , , with

(2.4)

Then problem (1.1) has at least one weak solution. Besides, any solution satisfies the estimate

(2.5)

Proof.

Let be different complete orthonormal systems for and set

(2.6)

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

We search for solutions of the approximate problem

(2.7)

To solve this algebraic system we define the operator

(2.8)

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

From the continuity of and , with respect to , we denote that is continuous. Therefore, from (H1), (H2), (H4) and Hölder's inequality, we note that

(2.9)

On the other hand, by Lemma 2.2, we have

(2.10)

where is constant.

From (2.9) and (2.10), we can prove that

(2.11)

This shows that there exists , depending only on , such that if . Then system (2.7) has a solution satisfying

(2.12)

From this bound estimate, going to a subsequence if necessary, there are and such that

(2.13)

Besides, since , compactly and the mapping is, respectively, continuous and

(2.14)

Then fixing in (2.7) and letting , we conclude that

(2.15)

From the completeness of , identity holds with replaced by any . In particularly, when , we get

(2.16)

On the other hand, let in (2.7) and passing to the limit, we get

(2.17)

Then we conclude that , which shows that is a solution of (1.1). Finally, if is any solution of (1.1) and is nontrivial, then

(2.18)

The proof is complete.

3. Variational Method

In this section, we consider the following problem:

(3.1)

where are constants, and 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 be a Banach space and let satisfy the Palais-Smale condition. Suppose also that

(i)

• (ii) there exist constants such that , if ,

• (iii) there exists an element with .

Define . Then

(3.2)

is a critical value of .

Theorem 3.2.

Assume the conditions (H1)–(H3) hold. In addition, the function satisfies

(H5)there exist with and , such that , , where .

Then (3.1) has a nontrivial solution.

Proof.

The weak solutions of (3.1) are critical points of the functional defined by

(3.3)

where .

Let us check the condition. Let , we have

(3.4)

Let be a Palais-Smale sequence in , that is, and and assume the contradiction that , then, from (H1), (H5), we have

(3.5)

where . Then by the Sobolev embedding theorem and Lemma 2.2, we can select such that

(3.6)

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

(3.7)

So we obtain

(3.8)

Let , we get

(3.9)

Let , then we take such that , when is sufficient small.

So for and small enough, then we have for all .

On the other hand, take with for , we have

(3.10)

Since , we obtain when .

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