Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space
J. M. Do Ó, R. F. Freire, J. Giacomoni, E. S. Medeiros
TL;DR
The paper analyzes a weighted elliptic problem in the half-space with a nonlinear boundary condition, showing that a spatially varying diffusion weight can reverse the classical existence/nonexistence pattern of the unweighted Laplacian. It develops a variational framework in the weighted Sobolev space $\mathcal{D}^{1,2}_\rho(\mathbb{R}^N_+)$, establishes sharp embeddings and attainability of best constants in critical regimes, and proves regularity results up to $C^{2,\alpha}_{\mathrm{loc}}$ (and $H^2_{\mathrm{loc}}$) for nonnegative weak solutions. A new Pohozaev-type identity is derived under suitable conditions on $\rho$, leading to precise nonexistence results in several parameter regimes and highlighting the inversion of classical solvability due to the weight. The combination of these analytical tools—Hardy–Sobolev-type embeddings, radial compactness, variational Mountain Pass methods, and Pohozaev identities—provides a comprehensive picture of existence, regularity, and nonexistence for weighted elliptic problems with boundary nonlinearities in the half-space.
Abstract
In this paper we study the problem $-\mathrm{div}(ρ(x_N)\nabla u)=a|u|^{p-2}u$ in $\mathbb{R}^N_+$, $-\partial u/\partial x_N=b|u|^{q-2}u$ in $\mathbb{R}^{N-1}$ where $a,b \in \mathbb{R}$, $p,q\in (1,\infty)$ 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 $-\mathrm{div}(ρ(x_N)\nabla u)$ 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.