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Zbl 0866.35039
Gui, Changfeng
Multipeak solutions for a semilinear Neumann problem.
(English)
[J] Duke Math. J. 84, No.3, 739-769 (1996). ISSN 0012-7094

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 $\bbfR^N$, $\nu$ 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 $a\ge 0$ 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 $\varepsilon \downarrow 0$.
[K.-i.Nagasaki (Minneapolis)]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J20 Second order elliptic equations, variational methods
35B25 Singular perturbations (PDE)

Keywords: local maxima of a solution; asymptotic behavior

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