Semilinear elliptic degenerate equations with critical exponent

2024/03/19 by Nguyễn Minh Trí, Tri, N. M., Dang Anh Tuan +1
Computer Science · Mathematics · #35B33 #35J61 #35J70 #46E30 #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2403.12828

openalex publication_date 2024/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation -(∂² u)/(∂ x²) -|x|^2k(∂² u)/(∂ y²)=|x|^2ku^p+f(x,y,u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ², Ω∩ \x=0\≠ ∅, k∈\mathbb N, f(x,y,0)=0, and p=(4+5k)/k is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17]

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