The H\'enon equation in Orlicz-Sobolev spaces

Abstract

In this paper, we consider the H\'enon problem in the setting of Orlicz-Sobolev spaces: \begin{equation*} \begin{cases} -\Delta_g u= |x|^\alpha h( u) \quad \text{in }B\\ u>0 \quad \text{in }B\\ u= 0 \quad \text{on }\partial B\\ \end{cases} \end{equation*}where $B$ is the unit ball in $\mathbb{R}^n$, $g=G'$, $h=H'$ are N-functions and the operator $-\Delta_g$ is the $g$-Laplacian. We show that the symmetric term $|x|^\alpha$, for $\alpha>0$, allows to have radial solutions even for supercritical $H$, generalizing results for the classical H\'enon equation. We also show that radial solutions are indeed bounded. Finally, we state a Pohozaev's identity in Orlicz-Sobolev spaces that we apply to get a range in $\alpha$ for which the problem has no bounded solutions.