Hénon elliptic equations in $\mathbb R^2$ with subcritical and critical exponential growth

Abstract

We study the Dirichlet problem in the unit ball $B_1$ of $\mathbb R^2$ for the Hénon-type equation of the form \begin{equation*} \begin{cases} -\Delta u =\lambda u+|x|^{\alpha}f(u) & \mbox{in } B_1, \\ \quad \ \ u = 0 & \mbox{on } \partial B_1, \end{cases} \end{equation*} where $f(t)$ is a $C^1$-function in the critical growth range motivated by the celebrated Trudinger-Moser inequality. Under suitable hypotheses on constant $\lambda$ and $f(t)$, by variational methods, we study the solvability of this problem in appropriate Sobolev s paces.