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 Schrödinger-Poisson equation $$\{\begin{array}{ll}-\Delta u+V(x)u+\lambda u+(|x{|}^{-1}\ast u^2)u=|u{|}^{p-2}u& \text{in}{\mathbb{R}}^3,\\ u>0,{\int }_{{\mathbb{R}}^3}{u}^{2}dx=a^2,& \end{array}$$ 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 $V(x)\ge 0$, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al. For $V(x)\le 0$, we directly construct an entire mountain-pass solution with positive energy.