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Zbl 0933.35070
Dancer, E.N.; Yan, Shusen
Multipeak solutions for a singularly perturbed Neumann problem. (English)
[J] Pac. J. Math. 189, No.2, 241-262 (1999). ISSN 0030-8730

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 $N=2$.

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B25 Singular perturbations (PDE)
35B05 General behavior of solutions of PDE
35A15 Variational methods (PDE)

Keywords: decomposition lemma; $k$-peak solutions; existence

Cited in: Zbl 0948.35055

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