Ground States And Multiple Solutions For Schrodinger Equation

As we know, the existence and multiplicity of solutions of nonlinear Schrodinger equation has been widely investigated in the literature over the past several decades, among which the assumptions on nonlinearity are becoming more and more mild. But there are only few papers dealt with the case that0is a boundary point of the spectrum which is very difficult because H1(RN) is no longer the working space on which the energy functional defines. Indeed, the working space for this case is only a Banach space, not a Hilbert space. Although, existence of nontrivial solutions of this problem has been investigated under stronger assumptions on nonlinear term, we still need to ask more that if there exists ground states solutions of Nehari—Pankov type without the strictly increasing assumption? On the other hand, can the severe restrictions on nonlinearity be weakened to a milder version? In this paper, we give a positive answer to these questions.In the present paper, we primarily consider the ground states solutions for nonlinear Schrodinger equation with spectrum point zero. For the case that the nonlinearity is superlinear, we developed a direct and simple approach to obtain the existence of ground state solution of Nehari—Pankov type. For the asymptotically linear case, we are also able to obtain the existence of least energy solution and ground state solution of Nehari—Pankov type under mild assumptions on nonlinear term. To the best of author’s knowledge, this seems to be the first result for such an asymptotically linear Schrodinger equation. In the last section of this paper, we also consider a class of semilinear Schrodinger equation with sign-changing potential and nonlinearity. The existence of infinitely many pairs of solutions are obtained with general assumptions on nonlinear term.