Application Of Critical Point Theory In Schr (o | ¨) Dinger-Poisson System And Kirchhoff Equation

In this paper, we apply critical point theory 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 first chapter, we study the existence of solutions for a class of Kirchhoff-Poisson equation by the Nehari manifold methods. Under a general 4-superlinear condition on the nonlinearity, we prove the existence of a ground state solution. If the nonlinearity 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’.In second chapter, we study a class of Schrodinger-poisson equations with potential vanishing at infinity. Using Nehari manifold and variational methods, we find a least sign-changing solution to this problem. Moreover, if the problem presents symmetry, we prove the existence of infinitely many nontrivial solutions. In our results the nonlinearity is only required to be continuous.