A system with weights and with critical Sobolev exponent

2021/10/27 by Asma Benhamida, Rejeb Hadiji, Benhamida, Asma +1
Mathematics · #35A01 #35A15 #35J57 #35J62 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2110.14640

openalex publication_date 2021/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the minimization problem : inf_ u ∈ H₀¹(Ω), v ∈ H₀¹(Ω),
‖ u ‖_L^q =1, ‖ v ‖_L^q = 1 [ (1)/(2) ∫_Ω a(x) \vert ∇ u(x) \vert²dx + (1)/(2) ∫_Ω b(x) \vert ∇ v (x)\vert²dx - λ∫_Ω u(x)v (x)dx ] where q=(2N)/(N-2), N ≥ 4, a and b are two continuous positive weight functions. We show the existence of solutions of the previous minimizing problem under some conditions on a, b, the dimension of the space and the parameter λ.

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