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On the effect of critical points of distance function in superlinear elliptic problems. (English) Zbl 0989.35054

Summary: We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions,

-ε 2 Δu+u=u p inΩu>0inΩu=0oru v=0inΩ,

where Ω is a bounded, smooth domain of N , N2, ε>0, 1<p<N+2 N-2 if N3 or p>1 if N=2 and ν is the unit outward normal at the boundary of Ω. We show that any “suitable” critical point x 0 of the distance function generates a family of single interior spike solutions, whose local maximum point tends to x 0 as ε tends to zero.


MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35B25Singular perturbations (PDE)
35J20Second order elliptic equations, variational methods