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Multiple interior peak solutions for some singularly perturbed Neumann problems. (English) Zbl 1061.35502
Summary: We consider the problem ϵ 2 Δu-u+f(u)=0, u>0, in Ω,u/ν=0 on , where Ω is a bounded smooth domain in N ,ϵ>0 is a small parameter, and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as ϵ approaches zero, at a critical point of the mean curvature function H(P),PΩ. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function d(P,Ω),PΩ. In this paper, we prove the existence of interior K-peak (K2) solutions at the local maximum points of the following function: ϕ(P 1 ,P 2 ,,P K )=min i,k,l=1,,K;kl (d(P i ,Ω),1 2|P k -P l |). We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function ϕ(P 1 ,,P K ) appears naturally in the asymptotic expansion of the energy functional.
MSC:
35J25Second order elliptic equations, boundary value problems
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)
47N20Applications of operator theory to differential and integral equations