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Zbl 0944.35021
Wei, Juncheng
On the interior spike solutions for some singular perturbation problems.
(English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 128, No.4, 849-874 (1998). ISSN 0308-2105; ISSN 1473-7124/e

The paper deals with positive solutions to singularly perturbed semilinear elliptic problems of the following form: $$- \varepsilon^2\Delta u=f(u) \quad \text{in } \Omega u=0 \quad \text{ or} \quad \frac{\partial u}{\partial \nu}=0 \quad\text{on }\partial \Omega .$$ Here $\Omega \subseteq \bbfR^n$ is a bounded domain with smooth boundary $\partial \Omega$ and $f:\bbfR \rightarrow \bbfR$ is smooth; a typical choice is $f(u)=-u+u^p (p>1)$. The purpose of the paper is to investigate the existence of single interior spike solutions. A family of solutions $\{u_\varepsilon\}$ is called single peaked if: $(i)$ the energy is bounded for any $\varepsilon>0$; $(ii)$ $u_\varepsilon$ has only one local maximum point $P_\varepsilon$, $P_\varepsilon \rightarrow P_0$, $u_\varepsilon \rightarrow 0$ in $C^1_{\text{loc}}(\overline \Omega \setminus P_0)$ and $u_\varepsilon(P_\varepsilon) \rightarrow \alpha>0$ as $\varepsilon \rightarrow 0$. The limiting point $P_0$ is called a boundary spike if $P_0 \in \partial \Omega$, respectively an interior spike if $P_0 \in \Omega$. In both cases the problem of existence and location of spikes has stimulated quite a few papers in recent years. The paper provides a unified approach to this problem under rather general assumptions on the function $f$ (both for Dirichlet and for Neumann homogeneous boundary conditions); in particular, necessary and sufficient conditions for the existence of interior spike solutions are given. A variety of methods is used to this purpose, including in particular weak convergence of measures and Lyapunov-Schmidt reduction.
[Alberto Tesei (Roma)]
MSC 2000:
*35J55 Systems of elliptic equations, boundary value problems
35B25 Singular perturbations (PDE)
35B65 Smoothness of solutions of PDE

Keywords: weak convergence of measures; Lyapunov-Schmidt reduction; single interior spike; boundary spike

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