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

TL;DR

The article analyzes a class of fractional $p$-Laplacian systems with weighted critical Hardy nonlinearities on bounded domains. It develops a variational framework to address both homogeneous and nonhomogeneous systems, establishing ground state existence in the subcritical weighted regime, a concentration-compactness principle to handle noncompactness at the weighted critical level, and existence of a least-energy nontrivial solution along with multiplicity results via Lusternik–Schnirelmann theory. The results give precise parameter thresholds (involving $\eta$, $\gamma$, and domain gravity through $\lambda_1$, $S_{\alpha+\beta}$, and $S_{\alpha,\beta}$) ensuring nontrivial solutions, and show that topology of $\Omega$ yields multiple solutions in the fully critical, weighted setting. These contributions advance the understanding of nonlocal elliptic systems with critical Hardy-type nonlinearities and provide a robust variational toolkit for related problems.

Abstract

Let $Ω\subset \mathbb{R}^d$ 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} &(-Δ_p)^s u= \fracα{q} \frac{|u|^{α-2}u|v|^β}{|x|^m} \;\;\text{in}\;Ω,\\ &(-Δ_p)^s v= \fracβ{q} \frac{|v|^{β-2}v|u|^α}{|x|^m}\;\;\text{in}\;Ω,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus Ω, \end{aligned} \right. \end{equation*} where $d>sp$, $α+ β= q$ 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} &(-Δ_p)^s u = η|u|^{r-2}u + γ\fracα{p_{s}^{*}(m)} \frac{|u|^{α-2}u|v|^β}{|x|^m} \;\;\text{in}\;Ω,\\ &(-Δ_p)^s v = η|v|^{r-2}v + γ\fracβ{p^{*}_{s}(m)} \frac{|v|^{β-2}v|u|^α}{|x|^m}\;\;\text{in}\;Ω,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus Ω, \end{aligned} \right. \end{equation*} where $η, γ> 0$ are parameters and $p \leq r < p_{s}^{*}(0)$. Depending on the values of $η, γ$, 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 $\text{cat}_Ω(Ω)$ nontrivial solutions.

Theorems & Definitions (51)

• Remark 1.1
• Definition 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3