p-Kirchhoff Modified Schrödinger Equation with Critical Nonlinearity in $\mathbb {R}^{N}$

Abstract

This article study the following on p-Kirchhoff modified Schrödinger equation with critical nonlinearity in $\mathbb {R}^{N}$:

$$\begin{aligned} {\mathcal {K}}(u)+V(x)|u{{|}^{p-2}}u= & {} \lambda \left( \int _{{{\mathbb {R}}^{N}}}{\frac{|u(y){{|}^{2p_{\mu }^{*}}}}{|y-x{{|}^{\mu }}}dy} \right) |u(x){{|}^{2p_{\mu }^{*}-2}}u(x)\\{} & {} +f(x,u) \text{ in }\ \mathbb {R}^{N}, \end{aligned}$$

where ${\mathcal {K}}(u)=-\left( a+b\int _{\mathbb {R}^{N}}|\nabla u|^{p}dx\right) \Delta _{p}u-au\Delta _{p}(u^{2})$ with $a>0, b\ge 0$, $0<\mu <N$, $N\ge 3$, and $\lambda >0$ is a positive parameter. Here $p^{*}_{\mu }:=\frac{p}{2}\frac{2N-\mu }{N-p}$ is the critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. First, we establish the concentration-compactness principle related to our problem. Moreover, under some suitable assumptions on the nonlinearity f and the potential V, the existence of infinitely many nontrivial solutions are obtained by using the concentration-compactness principle and the symmetric mountain pass theorem. We generalize and fill in some of the previous results (Liang et al., Math Methods Appl Sci 43:2473–2490, 2020; Liang and Song, Differ Integral Equ 35:359–370, 2022; Yang and Ding, Ann Mat Pura Appl 192:783–804, 2013).