Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition
© F.Wang and Y. An. 2009
Received: 15 December 2008
Accepted: 17 February 2009
Published: 8 March 2009
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.
where . It was proposed by Kirchhoff  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 . 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.
For convenience, we give the following hypotheses.
Lemma 2.2 (see ).
Assume that (H1)–(H3) hold. In addition, we suppose that
The proof is complete.
3. Variational Method
The nontrivial solution of (3.1) comes from the Mountain Pass Lemma in .
Lemma 3.1 (Mountain Pass Lemma).
Then (3.1) has a nontrivial solution.
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.
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