Existence and multiplicity results for nonlinear Schrödinger-Poisson equation with general potential

Abstract

This paper is concerned with the Schrödinger-Poisson equation

-\mathrm{\Delta }u+V(x)u+\phi (x)u=f(x,u),x\in {ℝ}^{\mathrm{3}},

-\mathrm{\Delta }\phi =u^{\mathrm{2}},\underset{\left|x\right|\to +\infty }{\lim }\phi (x)=\mathrm{0.}

Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of solutions are obtained via variational methods.

Key words： Schrdinger-Poisson equation; Morse index; variational method

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