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.