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The Dirichlet problem for singularly perturbed elliptic equations. (English) Zbl 0933.35083

This remarkable paper is devoted to the Dirichlet problem for a singularly perturbed elliptic equation

-ε 2 Δu ˜+u ˜=u ˜ q ,u ˜>0,

in a bounded domain Ω n , u ˜| Ω =0, where 1<q< if n{1,2} and 1<q<(n+2)/(n-2) if n3, ε>0 is a small real parameter. The authors present two main results concerning the existence of a family of solutions u ˜ ε of the problem under consideration. The first result is the following. Given the inequality max QV d(Q,Ω)<max QV ¯ d(Q,Ω), where d(Q,Ω)dist(Q,Ω), V is an open set and V ¯Ω. Then there exists ε ¯>0 and u ˜ ε for 0<ε<ε ¯ such that u ˜ ε has a unique local maximum point Q ˜ ε V, d(Q ˜ ε ,Ω)max QV ¯ d(Q,Ω) as ε0 and Q ˜ ε is the unique critical point of u ˜ ε provided that n{1,2} or Ω is convex. The second result consists in the following statement. If V is open in Ω, V ¯Ω, V𝒪 (𝒪Ω) and the Brouwer degree deg(d(Q,Ω),V,0)0, then there exists ε ¯>0 and u ˜ ε for 0<ε<ε ¯ such that u ˜ ε has a unique local maximum point Q ˜ ε V, d(Q ˜ ε ,S)0 (S=Ω𝒪) as ε0 and also Q ˜ ε is the unique critical point of u ˜ ε provided that n{1,2} or Ω is convex.


MSC:
35J70Degenerate elliptic equations
35B25Singular perturbations (PDE)
35B38Critical points in solutions of PDE