Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 257, pp. 1-8.
Title: Ground state solution of a nonlocal boundary-value problem
Author: Cyril Joel Batkam (Univ. de Sherbrooke, Quebec, Canada)
Abstract:
In this article, we apply the Nehari manifold method to study
the Kirchhoff type equation
$$
-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)
$$
subject to Dirichlet boundary conditions. Under a general 4-superlinear
condition on the nonlinearity f, we prove the existence of a ground
state solution, that is a nontrivial solution which has least energy
among the set of nontrivial solutions. If f is odd with respect
to the second variable, we also obtain the existence of infinitely
many solutions. Under our assumptions the Nehari manifold does not
need to be of class C^1.
Submitted June 24, 2013. Published November 22, 2013.
Math Subject Classifications: 35J60, 35J25.
Key Words: Nonlocal problem; Kirchhoff's equation; ground state solution;
Nehari manifold.