Concentration phenomena of normalized solutions for a fractional p-Laplacian Schrödinger–Choquard system in RN

Abstract

In this paper, we consider the following Schrödinger–Choquard system in $R$^N $\left(\begin{array}{c}{(-\Delta )}_p^s{u}_1+(V_1\left(\varepsilon x\right)-{\lambda }_1){\left|u_1\right|}^{p-2}{u}_1={\mu }_1[\frac{1}{{\left|x\right|}^{N-\alpha }}\ast F_1\left(u_1\right)]f_1\left(u_1\right)+\beta r_1{\left|u_1\right|}^{r_1-2}{u}_1{\left|u_2\right|}^{r_2}\text{in}{R}^N,\\ {(-\Delta )}_p^s{u}_2+(V_2\left(\varepsilon x\right)-{\lambda }_2){\left|u_2\right|}^{p-2}{u}_2={\mu }_2[\frac{1}{{\left|x\right|}^{N-\alpha }}\ast F_2\left(u_2\right)]f_2\left(u_2\right)+\beta r_2{\left|u_1\right|}^{r_1}{\left|u_2\right|}^{r_2-2}{u}_2\text{in}{R}^N,\\ {\int }_{R^N}{\left|u_1\right|}^{p}dx=a_1^p,{\int }_{R^N}{\left|u_2\right|}^{p}dx=a_2^p,\end{array}\right)$where ${(-\Delta )}_p^s$ is the fractional $p$-Laplacian operator, $\alpha \in (0,N)$, $s\in (0,1)$, $r_1,r_2>1$, $r_1+r_2\in (p,p+\frac{p^{2}s}{N})$, $\varepsilon >0$ is a parameter, ${\mu }_i,a_i>0$ $(i=1,2)$ and $\beta >0$ are prescribed, ${\lambda }_i$ are the Lagrange multipliers to be determined, the nonlinear functions $f_i$ are continuous, and the potential functions $V_i$ satisfy some suitable conditions. With the aid of the Ljusternik–Schnirelmann category theory and variational methods, we obtain the multiplicity and concentration phenomena of normalized solutions for the above system. As far as we know, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions for fractional $p$-Laplacian Schrödinger–Choquard system in $R^N$. To some extent, the main results included in this paper complement several recent contributions to the study of nonlinear Schrödinger systems (Chen and Zou, 2021; Gou and Jeanjean, 2016).