Normalized solutions of mass supercritical Schrodinger-Poisson equation with potential

Abstract

In this paper we prove the existence of normalized solutions $(\lambda,u)\subset (0,\infty)\times H^1(\mathbb{R}^3)$ to the following Schr\"{o}dinger-Poisson equation $$ \begin{cases} -\Delta u+V(x)u+\lambda u+(|x|^{-1}\ast u^2)u=|u|^{p-2}u&\text{in}\,\mathbb{R}^{3},\\ u>0,\quad \int_{\mathbb{R}^{3}}u^2dx=a^2, \end{cases} $$ where $a>0$ is fixed, $p\in(\frac{10}{3},6)$ is a given exponent and the potential $V$ satisfies some suitable conditions. Since the $L^2(\mathbb{R}^3)$-norm of $u$ is fixed, $\lambda$ appears as a Lagrange multiplier. For $V(x)\geq0$, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al. For $V(x)\leq0$, we directly construct an entire mountain-pass solution with positive energy.