Small normalised solutions for a Schr\"odinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

TL;DR

This paper studies solutions to a nonlinear Schr"odinger-Poisson system in expanding domains, showing multiplicity related to domain topology and convergence to a ground state as the domain grows infinitely large.

Contribution

It establishes the existence of multiple solutions based on domain topology and describes their asymptotic behavior in large domains.

Findings

Number of solutions at least equals the Ljusternick-Schnirelmann category of the domain.

Solutions converge to a ground state in a0space as domain expands.

Results hold for arbitrary large domain size and small mass a0.

Abstract

Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in } \lambda\Omega, -\Delta\phi =u^{2}& \quad \text{in }\lambda\Omega, u>0 &\quad \text{in }\lambda\Omega, u =\phi=0 &\quad \text{on }\partial (\lambda\Omega), \int_{\lambda\Omega}u^{2} \,\text{d} x=\rho^2 \end{array} \right. \end{equation*} in the expanding domain $\lambda\Omega\subset \mathbb R^{3}, \lambda>1$ and $p\in (2,3)$, in the unknowns $(u,\phi,\omega)$. We show that, for arbitrary large values of the expanding parameter $\lambda$ and arbitrary small values of the mass $\rho>0$, the number of solutions is at least the Ljusternick-Schnirelmann category of $\lambda\Omega$. Moreover we show that as $\lambda\to+\infty$ the solutions found converge to a ground state of the problem in the whole space $\mathbb R^{3}$.