p -Kirchhoff Modified Schrödinger Equation with Critical Nonlinearity in ℝ^N

This article study the following on p -Kirchhoff modified Schrödinger equation with critical nonlinearity in ℝ^N : 𝒦(u)+V(x)|u|^p-2u= λ( ∫ _ℝ^N|u(y)|^2p_μ^*/|y-x|^μdy) |u(x)|^2p_μ^*-2u(x) +f(x,u) in ℝ^N, where 𝒦(u)=-( a+b∫ _ℝ^N|∇ u|^pdx) Δ _pu-auΔ _p(u^2) with a>0, b≥ 0 , 0<μ 0 is a positive parameter. Here p^*_μ:=p/22N-μ/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).