Some existence results of solutions for modified Schrödinger-Poisson system with $p$-Laplacian and critical nonlinearity in $\mathbb R^3$

Abstract

This article deals with the existence of solutions for a class of modified Schrödinger-Poisson system with $p$-Laplacian and critical nonlinearity in $\mathbb{R}^{3}$. First, by using variable transformations, the quasilinear equation is reduced to a semilinear equation, whose corresponding functional is well defined in the classical Sobolev space and satisfies the geometric conditions of mountain pass theorem. Next, under suitable assumptions on potential functions and nonlinear terms, we can prove the compactness condition by using the concentration-compactness principle which is completely different from previous methods of proving compactness conditions. Finally, some existence results of solutions for this problem are obtained with the help of the variational method. In a sense, we generalize and fill in some of the previous results [17, 24, 32, 33].