Application Of Nehari Manifold In Nonlocal Elliptic Equations

In this paper, we apply Nehari manifold and critical point method to study two kinds of nonlocal elliptic partial differential equations aiming to establish the existence of solutions, multiplicity of solutions and existence of sign-changing solutions of those systems.In the first part we apply the methods of Nehari manifold to study a class of Schrodinger-Poisson system in a bounded domain in R3 submitted to Dirichlet boundary conditions. Under a general 4-superlinear conditions on the non-linearity of f, we prove the existence of a ground state solution. If f is odd with respect to the second variable, we obtain the existence of infinitely many solutions. In the proof, the Nehari manifold does not need to be of C class.In second part of the paper, we prove the existence of sign-changing solutions for the Kirchhoff- Poisson equation in a bounded domain in R3 via Nehari manifold and variational method. As far as we know, this is the first time that this problem is studied.We find out some mistakes in M. Giovany and R. G. Nascimento’s paper and correct them in this paper.