On Choquard-Kirchhoff Type Critical Multiphase Problem
Anupma Arora, Gaurav Dwivedi
TL;DR
This work develops a variational framework for a Choquard-Kirchhoff type multiphase problem with variable exponents and critical growth in Musielak–Orlicz spaces. By proving embedding results, a tailored concentration-compactness principle, and an HLS-type inequality in $W^{1,{\mathcal T}}(\Omega)$, the authors obtain two nontrivial weak solutions for small $\lambda$ and large $\kappa$, and, under alternate conditions, at least $n$ pairs of solutions via a genus-based, symmetric variational approach. The second main result extends to infinitely many solutions through genus theory and an even functional framework, under appropriate PS conditions. Overall, the paper advances the theory of nonlocal, multiphase operators with variable exponent growth and Choquard nonlinearities by providing rigorous existence and multiplicity results in a broad Musielak–Orlicz setting with critical nonlinearity.
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) & =λg(x)\lvert{u}\rvert^{γ(x)-2}u+θB(x,u)+κ\left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}\, dy\right) f(x,u) \ \text{in} \ Ω, & u=0 \ \text{on} \ {\partial Ω}. \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)$.