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Multipeak solutions for a singularly perturbed Neumann problem. (English) Zbl 0933.35070
Summary: The aim of this paper is to prove the existence of \(k\)-peak solutions (solutions with more than one local maximum point) for the following singularly perturbed problem without imposing any extra condition on the boundary \(\partial\Omega\): \[ \begin{cases} -\varepsilon^2\Delta u+u=u^{p-1},\quad & \text{in }\Omega\\ u>0,\quad & \text{in }\Omega\\ {\partial u\over \partial n}=0, \quad &\text{on } \partial\Omega\end{cases}\tag{1} \] where \(\varepsilon\) is a small positive number, \(\Omega\) is a bounded \(C^3\)-domain in \(\mathbb{R}^N\), \(n\) is the unit outward normal of \(\partial\Omega\) at \(y\), \(2<p<{2N\over N-2}\) if \(N\geq 3\) and \(2<p<+\infty\) if \(N=2\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A15 Variational methods applied to PDEs
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