Ground state solution of a nonlocal boundary-value problem

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.

Keywords

• Nonlocal problem
• Kirchhoff's equation
• ground state solution
• Nehari manifold