The planar Schrodinger-Poisson system with a positive potential

In this paper we consider the problem { -Delta u +/- phi u + W' (x, u) = 0 in R-2, Delta phi = u(2) in R-2, where W is assumed nonnegative. In dimension three, the problem with the sign +(we call it (P+)) was considered and solved in [22], whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign - (say it (P-)). We provide a general existence result for (P+) and two examples falling in the case (P-) for which there exists at least a nontrivial solution.

Automated summary

This paper investigates a problem in two dimensions, where the existence for the sign + has been proved and two examples have been provided for the sign -.