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

- Fanglei Wang
^{1}Email author and - Yukun An
^{1}

**Received: **15 December 2008

**Accepted: **17 February 2009

**Published: **8 March 2009

## 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

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

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].

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 is bounded.

For convenience, we give the following hypotheses.

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

(H3) The function is not identically zero.

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

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]).

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

Proof.

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

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

*Hölder's*inequality, we note that

The proof is complete.

## 3. Variational Method

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

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.

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.

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

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

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

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.

## Authors’ Affiliations

## References

- Kirchhoff G:
*Mechanik*. Teubner, Leipzig, Germany; 1883.Google Scholar - Lions J-L: On some questions in boundary value problems of mathematical physics. In
*Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Mathematics Studies*.*Volume 30*. Edited by: de la Penha G, Medeiros LA. North-Holland, Amsterdam, The Netherlands; 1978:284–346.Google Scholar - Arosio A, Panizzi S: On the well-posedness of the Kirchhoff string.
*Transactions of the American Mathematical Society*1996, 348(1):305–330. 10.1090/S0002-9947-96-01532-2MATHMathSciNetView ArticleGoogle Scholar - Ono K: On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation.
*Journal of Mathematical Analysis and Applications*1997, 216(1):321–342. 10.1006/jmaa.1997.5697MATHMathSciNetView ArticleGoogle Scholar - Aives CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type.
*Computers & Mathematics with Applications*2005, 49(1):85–93. 10.1016/j.camwa.2005.01.008MathSciNetView ArticleGoogle Scholar - Ma TF: Remarks on an elliptic equation of Kirchhoff type.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 63(5–7):e1967-e1977.MATHView ArticleGoogle Scholar - Perera K, Zhang Z: Nontrivial solutions of Kirchhoff-type problems via the Yang index.
*Journal of Differential Equations*2006, 221(1):246–255. 10.1016/j.jde.2005.03.006MATHMathSciNetView ArticleGoogle Scholar - Zhang Z, Perera K: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow.
*Journal of Mathematical Analysis and Applications*2006, 317(2):456–463. 10.1016/j.jmaa.2005.06.102MATHMathSciNetView ArticleGoogle Scholar - Stańczy R: Nonlocal elliptic equations.
*Nonlinear Analysis: Theory, Methods & Applications*2001, 47(5):3579–3584. 10.1016/S0362-546X(01)00478-3MATHMathSciNetView ArticleGoogle Scholar - Corrêa FJSA, de Morais Filho DC: On a class of nonlocal elliptic problems via Galerkin method.
*Journal of Mathematical Analysis and Applications*2005, 310(1):177–187. 10.1016/j.jmaa.2005.01.052MATHMathSciNetView ArticleGoogle Scholar - Bonder JF, Rossi JD: Existence results for the
-Laplacian with nonlinear boundary conditions.
*Journal of Mathematical Analysis and Applications*2001, 263(1):195–223. 10.1006/jmaa.2001.7609MATHMathSciNetView ArticleGoogle Scholar - Chaïb K: Necessary and sufficient conditions of existence for a system involving the
-Laplacian
.
*Journal of Differential Equations*2003, 189(2):513–525. 10.1016/S0022-0396(02)00094-3MATHMathSciNetView ArticleGoogle Scholar - Amster P, Mariani MC, Méndez O: Nonlinear boundary conditions for elliptic equations.
*Electronic Journal of Differential Equations*2005, 2005(144):1–8.Google Scholar - Song S-Z, Tang C-L: Resonance problems for the
-Laplacian with a nonlinear boundary condition.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(9):2007–2021. 10.1016/j.na.2005.07.035MATHMathSciNetView ArticleGoogle Scholar - Zhao J-H, Zhao P-H: Existence of infinitely many weak solutions for the
-Laplacian with nonlinear boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(4):1343–1355. 10.1016/j.na.2007.06.036MATHMathSciNetView ArticleGoogle Scholar - Adams RA, Fournier JJF:
*Sobolev Spaces*. Academic Press, Amsterdam, The Netherlands; 2003.MATHGoogle Scholar - Evans LC:
*Partial Differential Equations*. American Mathematical Society, Providence, RI, USA; 1998.MATHGoogle Scholar

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