*(English)*Zbl 0944.35021

The paper deals with positive solutions to singularly perturbed semilinear elliptic problems of the following form:

Here ${\Omega}\subseteq {\mathbb{R}}^{n}$ is a bounded domain with smooth boundary $\partial {\Omega}$ and $f:\mathbb{R}\to \mathbb{R}$ is smooth; a typical choice is $f\left(u\right)=-u+{u}^{p}(p>1)$. The purpose of the paper is to investigate the existence of single interior spike solutions. A family of solutions $\left\{{u}_{\epsilon}\right\}$ is called single peaked if: $\left(i\right)$ the energy is bounded for any $\epsilon >0$; $\left(ii\right)$ ${u}_{\epsilon}$ has only one local maximum point ${P}_{\epsilon}$, ${P}_{\epsilon}\to {P}_{0}$, ${u}_{\epsilon}\to 0$ in ${C}_{\text{loc}}^{1}(\overline{{\Omega}}\setminus {P}_{0})$ and ${u}_{\epsilon}\left({P}_{\epsilon}\right)\to \alpha >0$ as $\epsilon \to 0$. 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 (PDE) |

35B65 | Smoothness and regularity of solutions of PDE |