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-Schrödinger problem involving Stein-Weiss type nonlinearity: $$ \mathcal K(u)= \lambda f(x) |u(x)|^{q-2} u(x)+ \Big ( \int_{\mathbb R^N}\frac{|u(y)|^{2p_{\beta,\mu}^{*}}} {|x-y|^{\mu}|y|^{\beta}}dy \Big ) \frac {|u(x)|^{2p_{\beta,\mu}^{*}-2} u(x)}{|x|^\beta} \; \text{ in }\; \mathbb R^N, $$ where $\lambda > 0$ is a parameter, $N\geq 3$, $$ \mathcal K(u) = \Big ( a+b \int_{\mathbb R^N}|\nabla u|^{p}dx \Big ) \Delta_{p} u - a u \Delta_{p}(u^2) $$ with $a > 0$, $b\geq 0$, $\beta\geq0,$ $0 < \mu < N$, $0 < 2\beta+ \mu < N$, $2\leq q < 2 p^*$. Here, $p^*=\frac{Np}{N-p}$ is the Sobolev critical exponent and $ p_{\beta,\mu}^{*}:= \frac p2\frac{(2N-2\beta-\mu)}{N-p} $ 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 $\lambda$ by using Krasnoselskii's genus theory, symmetric mountain pass theorem and $\mathbb Z_2$- symmetric version of mountain pass theorem for different ranges of $q$. We further show that these solutions belong to $L^\infty(\mathbb R^N)$.