Multiple normalized solutions to the nonlinear Schrödinger–Poisson system with the $L^2$-subcritical growth

Abstract

In this paper, we study the existence of multiple normalized solutions to the following Schrödinger–Poisson system with general nonlinearities: \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+\phi u=f(u)+\lambda u & \hbox{in $\mathbb{R}^3$,} \\ -\varepsilon^2\Delta\phi=u^2& \hbox{in $\mathbb{R}^3$,}\\ \int_{\mathbb{R}^3}|u|^2{\rm d}x=\varepsilon^3 a^2,\ \end{cases} \end{equation*} where $\varepsilon$, $a>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V(x):\mathbb{R}^3 \rightarrow [0,\infty)$ is a continuous function, and $f$ is a differentiable function satisfying $L^2$-subcritical growth. Through using the minimization techniques and the Lusternik–Schnirelmann category, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V(x)$ attains its minimum value.

Keywords

• schrödinger–poisson system
• normalized solutions
• lusternik–schnirelmann category
• variational methods