Fractional $p$-Laplace systems with critical Hardy nonlinearities: Existence and Multiplicity

Abstract

Let $\Omega \subset \mathbb{R}^d$ be a bounded open set containing zero, $s \in (0,1)$ and $p \in (1, \infty)$. In this paper, we first deal with the existence, non-existence and some properties of ground-state solutions for the following class of fractional $p$-Laplace systems \begin{equation*} \left\{\begin{aligned} &(-\Delta_p)^s u= \frac{\alpha}{q} \frac{|u|^{\alpha-2}u|v|^{\beta}}{|x|^m} \;\;\text{in}\;\Omega,\\ &(-\Delta_p)^s v= \frac{\beta}{q} \frac{|v|^{\beta-2}v|u|^{\alpha}}{|x|^m}\;\;\text{in}\;\Omega,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus \Omega, \end{aligned} \right. \end{equation*} where $d>sp$, $\alpha + \beta = q$ where $p \leq q \leq p_{s}^{*}(m)$ where $p_{s}^{*}(m) = \frac{p(d-m)}{d-sp}$ with $0 \leq m \le sp$. Additionally, we establish a concentration-compactness principle related to this homogeneous system of equations. Next, the main objective of this paper is to study the following non-homogenous system of equations \begin{equation*} \left\{\begin{aligned} &(-\Delta_p)^s u = \eta |u|^{r-2}u + \gamma \frac{\alpha}{p_{s}^{*}(m)} \frac{|u|^{\alpha-2}u|v|^{\beta}}{|x|^m} \;\;\text{in}\;\Omega,\\ &(-\Delta_p)^s v = \eta |v|^{r-2}v + \gamma \frac{\beta}{p^{*}_{s}(m)} \frac{|v|^{\beta-2}v|u|^{\alpha}}{|x|^m}\;\;\text{in}\;\Omega,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus \Omega, \end{aligned} \right. \end{equation*} where $\eta, \gamma > 0$ are parameters and $p \leq r < p_{s}^{*}(0)$. Depending on the values of $\eta, \gamma$, we obtain the existence of a non semi-trivial solution with the least energy. Further, for $m=0$, we establish that the above problem admits at least $\text{cat}_{\Omega}({\Omega})$ nontrivial solutions.