On singular perturbation problems with Robin boundary condition
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 199-230

We consider the following singularly perturbed elliptic problem ϵ 2 Δu-u+f(u)=0,u>0inΩ,ϵu ν+λu=0onΩ, where f satisfies some growth conditions, 0λ+, and Ω N (N>1) is a smooth and bounded domain. The cases λ=0 (Neumann problem) and λ=+ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant λ * >1 such that, as ϵ0, the least energy solution has a spike near the boundary if λλ * , and has an interior spike near the innermost part of the domain if λ>λ * . Central to our study is the corresponding problem on the half space.

Classification:  35B35,  35J40,  92C40
@article{ASNSP_2003_5_2_1_199_0,
     author = {Berestycki, Henri and Wei, Juncheng},
     title = {On singular perturbation problems with Robin boundary condition},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     pages = {199-230},
     zbl = {1121.35008},
     mrnumber = {1990979},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0}
}
Berestycki, Henri; Wei, Juncheng. On singular perturbation problems with Robin boundary condition. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 199-230. http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/

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