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

Manuel del Pino·Claudio Muñoz
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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 $Ω$ is a bounded, smooth domain in $\R^2$, $ϕ_1$ is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and $h\in\mathcal{C}^{0,α}(\barΩ)$. 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 $ϕ_1$ as $s\to +\infty$.

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Manuel del Pino, Claudio Muñoz. 2006-08-07. The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity. https://arxiv.org/abs/math/0608168

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