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Multipeak solutions for a semilinear Neumann problem. (English) Zbl 0866.35039

The paper is concerned with the semilinear Neumann problem:

${\epsilon }^{2}{\Delta }u-u+f\left(u\right)=0,\phantom{\rule{4pt}{0ex}}u>0,\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial \nu }=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },$

where ${\Omega }$ is a bounded domain in ${ℝ}^{N}$, $\nu$ is the outer normal to $\partial {\Omega }$ and $\epsilon$ is a positive constant. In addition to suitable conditions on $f\left(t\right)$, typically satisfied by the function $f\left(t\right)={t}^{p}-a{t}^{q}$ if $a\ge 0$ and $1, the domain ${\Omega }$ is assumed to satisfy the condition that there exist $k$ disjoint patches ${{\Lambda }}_{1}$, ${{\Lambda }}_{2},\cdots ,{{\Lambda }}_{k}$ on $\partial {\Omega }$ such that ${max}_{P\in {{\Lambda }}_{i}}H\left(P\right)>{max}_{P\in \partial {{\Lambda }}_{i}}H\left(P\right)$, where $H\left(P\right)$ denotes the mean curvature of $\partial {\Omega }$ at $P$. Under these conditions, the author proves the existence of a classical solution ${u}_{\epsilon }$ which has exactly $k$ local maxima, precisely one on each ${{\Lambda }}_{i}\left(i=1,2,\cdots ,k\right)$, and then analyses the asymptotic behavior as $\epsilon ↓0$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods 35B25 Singular perturbations (PDE)
##### Keywords:
local maxima of a solution; asymptotic behavior