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 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  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.
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
where are constants, , .
(H3) The function is not identically zero.
Let be endowed with norm . Then is a Banach space.
for all .
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 ).
If , then the embedding still holds for . Moreover, if , then the embedding is compact.
Assume that (H1)–(H3) hold. In addition, we suppose that
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 .
where is constant.
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 .
Lemma 3.1 (Mountain Pass Lemma).
Let be a Banach space and let satisfy the Palais-Smale condition. Suppose also that
there exist constants such that , if ,
there exists an element with .
is a critical value of .
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.
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.
Let , then we take such that , when is sufficient small.
So for and small enough, then we have for all .
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.
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