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

2025/04/28 by Biswas, Nirjan, Das, Paramananda, Gupta, Shilpa
#35B33 #35J50 #35J60 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.19513

Abstract

Let Ω⊂ ℝ^d be a bounded open set containing zero, s ∈ (0,1) and p ∈ (1, ∞). 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 \\beginaligned amp;(-Δ_p)^s u= \fracαq \frac|u|^α-2u|v|^β|x|^m in Ω,
amp;(-Δ_p)^s v= \fracβq \frac|v|^β-2v|u|^α|x|^m in Ω,
amp;u=v=0 in ℝ^d∖ Ω, \endaligned . where d>sp, α+ β= q where p ≤ q ≤ p_s^*(m) where p_s^*(m) = (p(d-m))/(d-sp) with 0 ≤ m ≤ 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 \\beginaligned amp;(-Δ_p)^s u = η|u|^r-2u + γ\fracαp_s^*(m) \frac|u|^α-2u|v|^β|x|^m in Ω,
amp;(-Δ_p)^s v = η|v|^r-2v + γ\fracβp^*_s(m) \frac|v|^β-2v|u|^α|x|^m in Ω,
amp;u=v=0 in ℝ^d∖ Ω, \endaligned . where η, γ> 0 are parameters and p ≤ 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 cat_Ω(Ω) nontrivial solutions.

Citations

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• On p-fractional weakly-coupled system with critical nonlinearities
• On critically coupled (s₁, s₂)-fractional system of Schrödinger equations with Hardy potential