Multiple solutions for planar Schrodinger-Poisson system with logarithmic nonlinearity

Abstract

This paper is concerned with the following planar Schrodinger-Poisson system: $\begin{cases} -\Delta u +V(x)u+\lambda\phi u=u\ln u^2,&\text{in }{R}^2,\\ \Delta\phi=u^2,&\text{in }{R}^2. \end{cases}$ where $\lambda\in{R}$ is a parameter and $V\in C^1({R}^2,R}^{+})$ is a coercive potential. Due to the presence of the logarithmic nonlinearity, the manifold method developed by Ruiz [R] is not applicable here. We apply a general minimax principle to an auxiliary functional and prove the existence of a nontrivial solution. Then, by minimizing the energy functional over the set of all nontrivial solutions, we succeed in showing that the system admits a ground state solution. If, in addition, $V$ is radially symmetric, we obtain infinitely many non radial sign-changing solutions.