Infinitely many solutions for a class of critical Kirchhoff-type equations involving p-Laplacian operator

Abstract

In this paper, we investigate the following Kirchhoff-type equation

$$\begin{aligned} -M\left( \int \limits _{\mathbb {R}^{N}}|\nabla u|^{p}{\mathrm{d}}x\right) \Delta _{p} u=|u|^{p^{*}-2}u+h(x)|u|^{q-2}u, \ x\in \mathbb {R}^{N}, \end{aligned}$$

where $N\ge 3$, $1< p<N,\ p^*=\frac{Np}{N-p}$, $0< h\in L^\frac{p^*}{p^*-q}(\mathbb {R}^{N})$ with $q\in (1,p^*)$; M is a nonnegative continuous function with some growth conditions. We show that the above problem has infinitely many solutions by using variational methods.