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Zbl 1061.35502
Gui, Changfeng; Wei, Juncheng
Multiple interior peak solutions for some singularly perturbed Neumann problems.
(English)
[J] J. Differ. Equations 158, No. 1, 1-27 (1999). ISSN 0022-0396

Summary: We consider the problem $\epsilon^2\Delta u-u+f(u)=0,$ $u>0$, in $\Omega, \partial u/\partial\nu=0$ on $\partial$, where $\Omega$ is a bounded smooth domain in $\Bbb R^N, \epsilon>0$ is a small parameter, and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as $\epsilon$ approaches zero, at a critical point of the mean curvature function $H(P),\ P\in\partial\Omega$. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function $d(P,\partial\Omega), P\in\Omega$. In this paper, we prove the existence of interior $K$-peak $(K\geq2)$ solutions at the local maximum points of the following function: $\phi(P_1,P_2,\cdots,P_K)=\min_{i,k,l=1, \cdots,K;k\not=l}(d(P_i,\partial\Omega),{1\over2}|P_k-P_l|)$. We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function $\phi(P_1,\cdots,P_K)$ appears naturally in the asymptotic expansion of the energy functional.
MSC 2000:
*35J25 Second order elliptic equations, boundary value problems
35B25 Singular perturbations (PDE)
47J30 Variational methods
47N20 Appl. of operator theory to differential and integral equations

Keywords: multiple interior spikes; nonlinear elliptic equations

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