The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity

Abstract

We consider the problem of Ambrosetti-Prodi type \begin{equation}\label{0}\quad\begin{cases} Δu + e^u = sϕ_1 + h(x) &\hbox{in} Ω, u=0 & \hbox{on} \partial Ω, \end{cases} \nonumber \end{equation} where $\Omega$ is a bounded, smooth domain in ${\mathbb{R}}^2$, ${\varphi }_1$ is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and $h\in {\mathcal{C}}^{0,\alpha }(\bar{\Omega })$. We prove that given $k\ge 1$ this problem has at least $k$ solutions for all sufficiently large $s>0$, which answers affirmatively a conjecture by Lazer and McKenna \cite{LM1} for this case. The solutions found exhibit multiple concentration behavior around maxima of ${\varphi }_1$ as $s\to +\infty$.

24 pages, to appear in J. Diff. Eqns