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

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Research Article Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition Fanglei Wang and Yukun An Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Correspondence should be addressed to Fanglei Wang, [email protected] Received 15 December 2008; Accepted 17 February 2009 Recommended by Zhitao Zhang 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. Copyright q 2009 F. Wang and Y. An. 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.

1. Introduction In this paper, we deal with the following elliptic equation with nonlinear boundary condition: −Δu  u 

fx, u , M Ω |∇u|2  |u|2 dx 

∂u  gx, u, ∂γ

in Ω, 1.1

on ∂Ω,

where Ω is a bounded domain in RN with smooth boundary ∂Ω, N > 2, ∂/∂γ is the outer unite normal derivative, M : R → R is continuous, f : Ω × R → R, g : ∂Ω × R → R are Carath´eodory functions.   For 1.1, if the nonlocal term M Ω |∇u|2  |u|2 dx is replaced by M Ω |∇u|2 dx, then the equation  −M

 |∇u| dx Δu  fx, u, 2

Ω

in Ω

1.2

2

Boundary Value Problems

is related to the stationary analog of the Kirchhoff equation:  utt − M

 |∇x u| dx Δx u  fx, t, 2

Ω

1.3

where Ms  asb, a, b > 0. 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 M Ω |∇u|2 |u|2 dx is replaced by M Ω |u|2 dx, then the equation  −M

 |u| dx Δu  fx, u, 2

Ω

in Ω

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 auth

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