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.
Categories: math.AP