Normalized Solutions for the Choquard Equation With Potential: Concentration and Multiplicity

Abstract

This article focuses on the study of multiplicity and concentration behavior of normalized solutions for a Choquard equation with a local perturbation $$\begin{array}{r}\{\begin{array}{ll}-{\mathrm{\Delta }}_{p}u+V(ϵx)|u{|}^{p-2}u=\lambda |u{|}^{p-2}u+(I_{\alpha }*|u{|}^q)|u{|}^{q-2}u+\mu |u{|}^{s-2}u& \text{in }{\mathbb{R}}^N,\\ {\displaystyle {\int }_{{\mathbb{R}}^N}|u{|}^p\mathrm{d}x=a^p>0,}\end{array}\end{array}$$ where $a,ϵ>0$ , $2\le p<N$ , $(p(N+\alpha ))/2N<q<(p^2+p(N+\alpha ))/2N$ , $p<s<p+(p^2/N)$ , $\mu >0$ and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. Under natural hypotheses, combining the minimization techniques and Ljusternik–Schnirelmann category theory, we obtain the existence and concentration property of normalized solutions for $ϵ>0$ sufficiently small, as well as the multiplicity result depending on the topology of the set $M$ where the potential $V$ attains its global minimum, which indicates that the numbers of normalized solutions is determined by the topological structure of the set $M$ .