Abstract:In the present paper we are dealing with the following quasilinear elliptic problem:
\begin{equation*}
\begin{cases}
-\mathrm{div}(\rho(x_N) |\nabla u|^{p-2}\nabla u) =a|u|^{s-2}u &\mbox{in }&\ \mathbb{R}^N_+,
-|\nabla u|^{p-2}\frac{\partial u}{\partial x_N}=b|u|^{q-2}u&\mbox{on }&\
\mathbb{R}^{N-1},
\end{cases}\ \end{equation*} where $a,b\in \mathbb{R}$, $p,q,s\in(1,\infty)$ and $\rho$ is a continuous positive function on $[0,+\infty)$. We first prove new and sharp embedding results that we establish for the associted weighted energy spaces. In application, we establish existence and regularity of weak solutions to the above problem. We also prove for this problem the nonexistence of nontrivial weak solutions by a new Pohozaev-type identity we obtain. The new results about existence and nonexistence highlight the role of the weight $\rho$ on the solvability of the problem contrasting strongly with those when $\rho$ is constant.
From: Loïc Constantin [view email]
[v1]
Wed, 10 Jun 2026 11:32:03 UTC (42 KB)
[v2]
Wed, 22 Jul 2026 14:50:04 UTC (43 KB)
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