Nontrivial solutions of quasilinear Choquard equation involving the $p$-Laplacian operator and critical nonlinearities

Abstract

In this paper, we intend to study the following quasilinear Choquard equation involving the $p$-Laplacian operator and critical nonlinearities in $\mathbb{R}^N$: $$ -\big(\Delta_p u +\Delta_p (u^2)\big) = \lambda\big(I_\mu*|u|^q\big)|u|^{q-2}u + |u|^{2p^\ast-2}u \quad \mbox{in}\ \mathbb{R}^N, $$ where $\Delta_p u = \mathop{\rm div}(|\nabla u |^{p-2} \nabla u)$ is the $p$-Laplacian operator with $1 < p\leq N$, $N \geq 3$, $I_\mu(x)=|x|^{-\mu}$, $0 < \mu < N$, $\frac{p(N+\mu)}{N} \leq q < \frac{p(N+\mu)}{N-p}$, $p^*=\frac{Np}{N-p}$ is the critical exponent in the sense of Sobolev inequality, $\lambda$ is a positive parameter. The existence of nontrivial solutions are obtained by using the mountain pass theorem. The main features and novelty of the paper are the presence of the convolution terms as well as critical nonlinearities.