Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space

2025/10/07 by J. M. Ó, Ó, J. M. Do, Ricardo Freire +5
Mathematics · #Differential Equations and Boundary Problems #Numerical methods in inverse problems #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2510.05999

Abstract

In this paper we study the problem -div(ρ(x_N)∇ u)=a|u|^p-2u in ℝ^N₊, -∂ u/∂ x_N=b|u|^q-2u in ℝ^N-1 where a,b ∈ ℝ, p,q∈ (1,∞) and ρ is a positive weight. We establish regularity results for weak solutions and, using a variational approach combined with a new Pohozaev-type identity, we show that the introduction of the weighted operator -div(ρ(x_N)∇ u) can reverse the known solvability behavior of the classical Laplacian case. Specifically, we identify regimes where the problem admits solutions despite nonexistence for the corresponding case with -Δ, and vice versa, thus inverting the classical existence and nonexistence results.

Citations

• A Berestycki-Lions type result for a class of degenerate elliptic problems involving the Grushin Operator
• Multiscale modeling of diffusion in a crowded environment
• A critical fractional equation with concave-convex power nonlinearities
• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
• Uniqueness theorems through the method of moving spheres
• Boundary regularity for solutions of degenerate elliptic equations
• Nonlinear scalar field equations, I existence of a ground state
• Multidimensional nonlinear diffusion arising in population genetics

Related

• Trace and Hardy-Sobolev type inequalities and applications to a quasilinear elliptic problem in half-space
• On stable entire solutions of semi-linear elliptic equations with weights
• Properties of solutions to some weighted p-Laplacian equation
• On problems with weighted elliptic operator and general growth nonlinearities
• Nonlinear elliptic equations on the upper half space