Hénon type equations with one-sided exponential growth

Abstract

We deal with the following class of problems: \begin{equation*} \begin{cases} -\Delta u=\lambda u+|x|^{\alpha}g(u_+)+ f(x)&\mbox{in } B_1,\\ u =0&\mbox{on }\partial B_1, \end{cases} \end{equation*} where $B_1$ is the unit ball in $\mathbb R^2$, $g$ is a $C^1$-function in $[0,+\infty)$ which is assumed to be in the subcritical or critical growth range of Trudinger-Moser type and $f\in L^{\mu}(B_1)$ for some $\mu>2$. Under suitable hypotheses on the constant $\lambda$, we prove existence of at least two solutions to this problem using variational methods. In case of $f$ radially symmetric, the two solutions are radially symmetric as well.