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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:

-ε 2 Δu=f(u)inΩu=0oru ν=0onΩ·

Here Ω n is a bounded domain with smooth boundary Ω and f: 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 ε } is called single peaked if: (i) the energy is bounded for any ε>0; (ii) u ε has only one local maximum point P ε , P ε P 0 , u ε 0 in C loc 1 (Ω ¯P 0 ) and u ε (P ε )α>0 as ε0. The limiting point P 0 is called a boundary spike if P 0 Ω, respectively an interior spike if P 0 Ω. 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:
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B25Singular perturbations (PDE)
35B65Smoothness and regularity of solutions of PDE