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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.
35J25Second order elliptic equations, boundary value problems
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)
47N20Applications of operator theory to differential and integral equations