Multiplicity results for $p$-Kirchhoff modified Schrödinger equations with Stein-Weiss type critical nonlinearity in $\mathbb R^N$

Abstract

. In this article, we consider the following modified quasilinear critical Kirchhoff-Schr¨odinger problem involving Stein-Weiss type nonlinearity: where λ > 0 is a parameter, N = 0 < µ < N , 0 < 2 β + µ < N , 2 ≤ q < 2 p ∗ . Here p ∗ = NpN − p is the Sobolev critical exponent and p ∗ β,µ := p 2 (2 N − 2 β − µ ) N − 2 is the critical exponent with respect to the doubly weighted Hardy-Littlewood-Sobolev inequality (also called Stein- Weiss type inequality). Then by establishing a concentration-compactness argument for our problem, we show the existence of infinitely many nontrivial solutions to the equations with respect to the parameter λ by using Krasnoselskii’s genus theory, symmetric mountain pass theorem and Z 2 - symmetric version of mountain pass theorem for different range of q . We further show that these solutions belong to L ∞ ( R N ).