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

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## 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## 1. Introduction

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.

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].

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.

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.

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]).

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

Proof.

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 .

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 .

*Hölder's*inequality, we note that Open image in new window

where Open image in new window is constant.

The proof is complete.

## 3. Variational Method

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

- (ii)
there exist constants Open image in new window such that Open image in new window , if Open image in new window ,

- (iii)
there exists an element Open image in new window with Open image in new window .

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.

where Open image in new window .

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.

*Hölder*'s inequality, Sobolev embedding theorem, and Lemma 2.2, we have

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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