## 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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### References:

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