Positive singular solutions of a certain elliptic PDE

2025/03/12 by Mohammadnejad, Negar
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.09818

Abstract

In this paper, we investigate the existence of positive singular solutions for a system of partial differential equations on a bounded domain \ -Δu = (1+κ₁(x)) | ∇ v |^p · amp; in~~ B₁ \backslash \0\,
-Δv = (1+κ₂(x)) | ∇ u |^p · amp; in~~ B₁ \backslash \0\,
u = v = 0 · amp; on~~ ∂ B₁. . We investigate the existence of positive singular solutions within B₁, the unit ball centered at the origin in ℝ^N , under the conditions N ≥ 3 and (N)/(N-1) < p < 2 . Additionally, we assume that κ₁ and κ₂ are non-negative, continuous functions satisfying κ₁(0) = κ₂(0) = 0 . Our system is an extension of the PDE equation studied by Aghajani et al. (2021) under similar assumptions.

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