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On the interior spike solutions for some singular perturbation problems. (English) Zbl 0944.35021

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 \( \Omega \subseteq \mathbb{R}^n \) is a bounded domain with smooth boundary \(\partial \Omega\) and \(f:\mathbb{R} \rightarrow \mathbb{R}\) 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.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B25 Singular perturbations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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