Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations (Q2866545)

The authors consider electrostatic solutions of the Schrödinger-Poisson equation of the form NEWLINE\[NEWLINE\begin{aligned} &-\Delta u+\lambda u+\phi u=Q(x)|u|^{p-2}u,\\ &-\Delta\phi=u^2 \end{aligned} NEWLINE\]NEWLINE in \(\mathbb R^3\). By using variational methods, the existence of positive solutions is proved when \(4\leq p<6\), and it is shown that the number of positive solutions depends on the profile of \(Q\) such as the number of strict maximum points and the limit at infinity. Non-existence results are also obtained when \(2<p\leq 3\).