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

In this paper we study the problem $-\mathrm{d}\mathrm{i}\mathrm{v}(\rho (x_N)\mathrm{\nabla }u)=a\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in ${\mathbb{R}}_{+}^N$, $-\mathrm{\partial }u\mathrm{/}\mathrm{\partial }{x}_N=b\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$ in ${\mathbb{R}}^{N-1}$ where $a,b\in \mathbb{R}$, $p,q\in (1,\mathrm{\infty })$ and $\rho$ 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 $-\mathrm{d}\mathrm{i}\mathrm{v}(\rho (x_N)\mathrm{\nabla }u)$ 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 $-\mathrm{\Delta }$, and vice versa, thus inverting the classical existence and nonexistence results.