Solutions with prescribed mass for the p-Laplacian Schrödinger-Poisson system with critical growth

Abstract

In this paper, we focus on the existence and multiplicity of solutions for the p-Laplacian Schrödinger-Poisson system$\{\begin{array}{cc}-{\Delta }_{p}u+\gamma \varphi {\left|u\right|}^{p-2}u=\lambda {\left|u\right|}^{p-2}u+\mu |u{|}^{q-2}u+{\left|u\right|}^{p^{⁎}-2}u,& \text{in}{R}^3,\\ -\Delta \varphi ={\left|u\right|}^p,& \text{in}{R}^3,\end{array}$ with a prescribed mass given by$\underset{R^3}{\int }|u{|}^{p}dx=a^p,$ in the Sobolev critical case, where, $1<p<3,a>0$, and $\gamma >0$, $\mu >0$ are parameters, $p^{⁎}=\frac{3p}{3-p}$ is the Sobolev critical exponent, and $\lambda \in R$ is an undetermined parameter, acting as a Lagrange multiplier. We investigate this system under the $L^p$-subcritical perturbation $\mu |u{|}^{q-2}u$, with $q\in (p,p+\frac{p^2}{3})$, and establish the existence of multiple normalized solutions using the truncation technique, concentration-compactness principle, and genus theory. In the $L^p$-supercritical regime: $q\in (p+\frac{p^2}{3},p^{⁎})$, we prove two existence results for normalized solutions under different assumptions for the parameters $\gamma ,\mu$, by employing the Pohozaev manifold analysis, concentration-compactness principle and mountain pass theorem. This study presents new contributions regarding the existence and multiplicity of normalized solutions of the p-Laplacian critical Schrödinger-Poisson problem, perturbed with a subcritical term in the whole space $R^3$, for the first time.