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Multipeak solutions for a semilinear Neumann problem. (English) Zbl 0866.35039

The paper is concerned with the semilinear Neumann problem:

ε 2 Δu-u+f(u)=0,u>0,inΩ,u ν=0onΩ,

where Ω is a bounded domain in N , ν is the outer normal to Ω and ε is a positive constant. In addition to suitable conditions on f(t), typically satisfied by the function f(t)=t p -at q if a0 and 1<q<p<(N+2)/(N-2), the domain Ω is assumed to satisfy the condition that there exist k disjoint patches Λ 1 , Λ 2 ,,Λ k on Ω such that max PΛ i H(P)>max PΛ i H(P), where H(P) denotes the mean curvature of Ω at P. Under these conditions, the author proves the existence of a classical solution u ε which has exactly k local maxima, precisely one on each Λ i (i=1,2,,k), and then analyses the asymptotic behavior as ε0.

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