Small normalised solutions for a Schrödinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

Abstract

Given a smooth bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$, we consider the following nonlinear Schrödinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -Δu+ ϕu -\abs{u}^{p-2}u = ωu & \quad \text{in } λΩ, -Δϕ=u^{2}& \quad \text{in }λΩ, u>0 &\quad \text{in }λΩ, u =ϕ=0 &\quad \text{on }\partial (λΩ), \int_{λΩ}u^{2} \,\text{d} x=ρ^2 \end{array} \right. \end{equation*} in the expanding domain $\hat{I}»\hat{I}\text{c◯}\subset {\mathbb{R}}^3,\hat{I}»>1$ and $p\in (2,3)$, in the unknowns $(u,\ddot{I}•,\ddot{I}‰)$. We show that, for arbitrary large values of the expanding parameter $\hat{I}»$ and arbitrary small values of the mass $\ddot{I}>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $\hat{I}»\hat{I}\text{c◯}$. Moreover we show that as $\hat{I}»\to +\infty$ the solutions found converge to a ground state of the problem in the whole space ${\mathbb{R}}^3$.

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