On singular perturbation problems with Robin boundary condition
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 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},
     pages = {199--230},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     mrnumber = {1990979},
     zbl = {1121.35008},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/}
}
TY  - JOUR
AU  - Berestycki, Henri
AU  - Wei, Juncheng
TI  - On singular perturbation problems with Robin boundary condition
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2003
SP  - 199
EP  - 230
VL  - 2
IS  - 1
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/
LA  - en
ID  - ASNSP_2003_5_2_1_199_0
ER  - 
%0 Journal Article
%A Berestycki, Henri
%A Wei, Juncheng
%T On singular perturbation problems with Robin boundary condition
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2003
%P 199-230
%V 2
%N 1
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/
%G en
%F 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, Série 5, Tome 2 (2003) no. 1, pp. 199-230. http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/

[1] P. Bates - G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns. 160 (2000), 283-356. | MR | Zbl

[2] P. Bates - E. N. Dancer - J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Eqns. 4 (1999), 1-69. | MR | Zbl

[3] C. C. Chen - C. S. Lin, Uniqueness of the ground state solution of Δu+f(u)=0 in N ,N3, Comm. PDE. 16 (1991), 1549-1572. | MR | Zbl

[4] M. Del Pino - P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), 883-898. | MR | Zbl

[5] M. Del Pino - P. Felmer - J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), 63-79. | MR | Zbl

[6] M. Del Pino - P. Felmer - J. Wei, On the role of distance function in some singularly perturbed problems, Comm. PDE 25 (2000), 155-177. | MR | Zbl

[7] A. Dillon - P. K. Maini - H. G. Othmer, Pattern formation in generalized Turing systems, I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol. 32 (1994), 345-393. | MR | Zbl

[8] M. Del Pino - P. Felmer - J. Wei, Mutiple peak solutions for some singular perturbation problems, Cal. Var. PDE 10 (2000), 119-134. | MR | Zbl

[9] E. N. Dancer - J. Wei, On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqns. 157 (1999), 82-101. | MR | Zbl

[10] E. N. Dancer - S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math. 189 (1999), 241-262. | MR | Zbl

[11] M. J. Esteban - P. L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 1-14. | MR | Zbl

[12] R. Gardner - L. A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 53-72. | MR | Zbl

[13] C. Gui - J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Diff. Eqns. 158 (1999), 1-27. | MR | Zbl

[14] C. Gui - J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math. 52 (2000), 522-538. | MR | Zbl

[15] C. Gui - J. Wei - M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 249-289. | EuDML | Numdam | MR | Zbl

[16] B. Gidas - W.-M. Ni - L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R n , In: “Mathematical Analysis and Applications, Part A”, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, pp. 369-402. | MR | Zbl

[17] M. Grossi - A. Pistoia - J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Cal. Var. PDE 11 (2000) 143-175. | MR | Zbl

[18] M. K. Kwong, Uniqueness of positive solutions of Δu-u+u p =0 in R n , Arch. Rational Mech. Anal. 105 (1989), 243-266. | MR | Zbl

[19] Y.-Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. PDE 23 (1998), 487-545. | MR | Zbl

[20] Y.-Y. Li - L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), 1445-1490. | MR | Zbl

[21] P.L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, I., Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109-145. | EuDML | Numdam | MR | Zbl

[22] P. L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, II., Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223-283. | EuDML | Numdam | MR | Zbl

[23] P. L. Lions, “Generalized solutions of Hamilton-Jacobi equations”, Pitman, 1982. | MR | Zbl

[24] C.-S. Lin - W.-M. Ni - I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns. 72 (1988), 1-27. | MR | Zbl

[25] W.-M. Ni, Diffusion, cross-diffusion, and their spike- layer steady states, Notices of Amer. Math. Soc. 45 (1998), 9-18. | MR | Zbl

[26] W.-M. Ni - I. Takagi, On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991), 819-851. | MR | Zbl

[27] W.-M. Ni - I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281. | MR | Zbl

[28] W.-M. Ni - I. Takagi, Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Industrial Appl. Math. 12 (1995), 327-365. | MR | Zbl

[29] W.-M. Ni - I. Takagi - J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions, Duke Math. J. 94 (1998), 597-618. | MR | Zbl

[30] W.-M. Ni - J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731-768. | MR | Zbl

[31] J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqns. 129 (1996), 315-333. | MR | Zbl

[32] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns. 134 (1997), 104-133. | MR | Zbl

[33] J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. 50 (1998), 159-178. | MR | Zbl

[34] J. Wei, On the effect of the domain geometry in a singularly perturbed Dirichlet problem, Diff. Int. Eqns. 13 (2000), 15-45. | MR | Zbl

[35] J. Wei - M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 459-492. | EuDML | Numdam | MR | Zbl

[36] J. Wei - M. Winter, Multiple boundary spike solutions for a wide class of singular perturbation problems, J. London Math. Soc. 59 (1999), 585-606. | MR | Zbl