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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 $\partial\Omega$: $$\cases -\varepsilon^2\Delta u+u=u^{p-1},\quad & \text{in }\Omega\\ u>0,\quad & \text{in }\Omega\\ {\partial u\over \partial n}=0, \quad &\text{on } \partial\Omega\endcases\tag 1$$ where $\varepsilon$ is a small positive number, $\Omega$ is a bounded $C^3$-domain in $\bbfR^N$, $n$ is the unit outward normal of $\partial\Omega$ at $y$, $2<p<{2N\over N-2}$ if $N\ge 3$ and $2<p<+\infty$ if $N=2$.

35J65Nonlinear boundary value problems for linear elliptic equations
35B25Singular perturbations (PDE)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35A15Variational methods (PDE)
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