Mathematics – Analysis of PDEs
Scientific paper
2006-08-07
Mathematics
Analysis of PDEs
24 pages, to appear in J. Diff. Eqns
Scientific paper
We consider the problem of Ambrosetti-Prodi type \begin{equation}\label{0}\quad\begin{cases} \Delta u + e^u = s\phi_1 + h(x) &\hbox{in} \Omega, u=0 & \hbox{on} \partial \Omega, \end{cases} \nonumber \end{equation} where $\Omega$ is a bounded, smooth domain in $\R^2$, $\phi_1$ 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 $\phi_1$ as $s\to +\infty$.
Muñoz Claudio
Pino Manuel del
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