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Multipeak solutions for a semilinear Neumann problem. (English) Zbl 0866.35039
The paper is concerned with the semilinear Neumann problem: $\varepsilon^2 \Delta u-u+f(u) =0,\;u>0, \text{ in } \Omega,\quad {\partial u\over \partial\nu} =0\text{ on } \partial \Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$, $$\nu$$ is the outer normal to $$\partial\Omega$$ and $$\varepsilon$$ is a positive constant. In addition to suitable conditions on $$f(t)$$, typically satisfied by the function $$f(t)= t^p-at^q$$ if $$a\geq 0$$ and $$1<q<p <(N+2)/(N-2)$$, the domain $$\Omega$$ is assumed to satisfy the condition that there exist $$k$$ disjoint patches $$\Lambda_1$$, $$\Lambda_2, \dots, \Lambda_k$$ on $$\partial \Omega$$ such that $$\max_{P\in \Lambda_i} H(P)> \max_{P\in\partial \Lambda_i} H(P)$$, where $$H(P)$$ denotes the mean curvature of $$\partial \Omega$$ at $$P$$. Under these conditions, the author proves the existence of a classical solution $$u_\varepsilon$$ which has exactly $$k$$ local maxima, precisely one on each $$\Lambda_i (i=1,2, \dots,k)$$, and then analyses the asymptotic behavior as $$\varepsilon \downarrow 0$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B25 Singular perturbations in context of PDEs
##### Keywords:
local maxima of a solution; asymptotic behavior
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##### References:
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