On Choquard-Kirchhoff Type Critical Multiphase Problem

TL;DR

This paper proves the existence of solutions for a complex nonlinear PDE involving variable exponents, nonlocal terms, and critical growth, using advanced functional analysis techniques in Musielak-Orlicz Sobolev spaces.

Contribution

It introduces new existence and multiplicity results for a Choquard-Kirchhoff type problem with critical growth in Musielak-Orlicz Sobolev spaces, extending previous work to variable exponent and nonlocal contexts.

Findings

Existence of weak solutions established.

Multiplicity results demonstrated.

New embedding and concentration compactness principles developed.

Abstract

In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc} &-M(\varphi_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla u}\rvert^{p(x)-2}\nabla u+a_1(x)\lvert{\nabla u}\rvert^{q(x)-2}\nabla u+a_2(x)\lvert{\nabla u}\rvert^{r(x)-2}\nabla u) & =\lambda g(x)\lvert{u}\rvert^{\gamma(x)-2}u+\theta B(x,u)+\kappa \left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}\, dy\right) f(x,u) \ \text{in} \ \Omega, & u=0 \ \text{on} \ {\partial \Omega}. \end{array}\right. \end{equation*} The term $B(x,u)$ on the right-hand side generalizes the critical growth. We obtain existence and multiplicity results by establishing certain embedding results and concentration compactness principle along with the Hardy-Littlewood-Sobolev type inequality for the Musielak Orlicz Sobolev space $ W^{1,\mathcal{T}}(\q)$.