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On the boundary spike layer solutions to a singularly perturbed Neumann problem. (English) Zbl 0873.35007

This paper deals with the Neumann problem

ε 2 Δu-u+u p =0,u>0inΩ,u ν=0onΩ,(1)

where Ω n is a smooth bounded domain, 1<p<(n+2)/(n-2) when n3 and 1<p< when n=1,2. Associated with (1) is the functional

vI(ε,Ω,v)=1 2 Ω (ε 2 |v| 2 +v 2 )-1 p+1 Ω v p+1 ·

Solutions u ε of (1) are called single boundary peaked if lim ε0 ε -n I ε (u ε )=1 2I(w), with I ε (u ε )=I(ε,Ω,u ε ), I(w)=I(1, n ,w), where w is the positive radial solution of Δw-w+w p =0 in n , w(z)0 as |z|, w(0)=max z n w(z).

The author states the theorem: if u ε is a family of single boundary peaked solutions of (1), then, as ε0, u ε has only one local maximum point P ε and P ε Ω. Moreover, the tangential derivative of the mean curvature of Ω at P ε tends to zero. A converse of this theorem is also investigated. The particular case of the least-energy solutions of (1) was studied previously by W-M. Ni and I. Takagi [Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042), and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].

The proof is based on a decomposition of u ε of the form u ε =α ε w ε +v ε , where w ε is the solution of ε 2 Δu-u+w p ((x-P ε )/ε)=0 in Ω and u/ν=0 on Ω, and on fine estimates for α ε + and the error term v ε H 1 (Ω).

Reviewer: D.Huet (Nancy)
MSC:
35B25Singular perturbations (PDE)
35J65Nonlinear boundary value problems for linear elliptic equations
35B40Asymptotic behavior of solutions of PDE