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

Abstract

In this paper we study the problem $-\mathrm{div}(\ddot{I}(x_N)\nabla u)=a|u{|}^{p-2}u$ 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 $\ddot{I}$ 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 $-\mathrm{div}(\ddot{I}(x_N)\nabla u)$ 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 $-\hat{I}”$, and vice versa, thus inverting the classical existence and nonexistence results.