Ground state solution of a nonlocal boundary-value problem

Abstract

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=f(x,u) \end{equation*} submitted 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. In case which $f$ 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 ${\mathcal{C}}^1$.

8 pages

References in corpus (1)

• The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem

Cited by in corpus (2)

• Multiple sign-changing solutions to a class of Kirchhoff type problems
• Ground states of elliptic problems involving non homogeneous operators