On singular perturbation problems with Robin boundary condition

Henri Berestycki; Juncheng Wei

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 1, page 199-230
  • ISSN: 0391-173X

Abstract

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We consider the following singularly perturbed elliptic problem ϵ 2 Δ u - u + f ( u ) = 0 , u > 0 in Ω , ϵ ∂ u ∂ ν + λ u = 0 on ∂ Ω , where f satisfies some growth conditions, 0 ≤ λ ≤ + ∞ , and Ω ⊂ ℝ 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 ϵ → 0 , the least energy solution has a spike near the boundary if λ ≤ λ * , 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.

How to cite

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Berestycki, Henri, and Wei, Juncheng. "On singular perturbation problems with Robin boundary condition." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.1 (2003): 199-230. <http://eudml.org/doc/84496>.

@article{Berestycki2003,
abstract = {We consider the following singularly perturbed elliptic problem\[ \begin\{aligned\} \epsilon ^2 \Delta u -u + f(u)&=0, \ u&gt;0 \ \text\{in\} \ \Omega ,\\ \epsilon \frac\{\partial u\}\{\partial \nu \} + \lambda u &=0 \ \text\{on\} \ \partial \Omega , \end\{aligned\} \]where $f$ satisfies some growth conditions, $ 0 \le \lambda \le +\infty $, and $\Omega \subset \mathbb \{R\}^N$ ($N&gt;1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda = +\infty $ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant $\lambda _\{*\} &gt;1$ such that, as $\epsilon \rightarrow 0$, the least energy solution has a spike near the boundary if $\lambda \le \lambda _\{*\} $, and has an interior spike near the innermost part of the domain if $\lambda &gt; \lambda _\{*\} $. Central to our study is the corresponding problem on the half space.},
author = {Berestycki, Henri, Wei, Juncheng},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {199-230},
publisher = {Scuola normale superiore},
title = {On singular perturbation problems with Robin boundary condition},
url = {http://eudml.org/doc/84496},
volume = {2},
year = {2003},
}

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
PB - Scuola normale superiore
VL - 2
IS - 1
SP - 199
EP - 230
AB - We consider the following singularly perturbed elliptic problem\[ \begin{aligned} \epsilon ^2 \Delta u -u + f(u)&=0, \ u&gt;0 \ \text{in} \ \Omega ,\\ \epsilon \frac{\partial u}{\partial \nu } + \lambda u &=0 \ \text{on} \ \partial \Omega , \end{aligned} \]where $f$ satisfies some growth conditions, $ 0 \le \lambda \le +\infty $, and $\Omega \subset \mathbb {R}^N$ ($N&gt;1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda = +\infty $ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant $\lambda _{*} &gt;1$ such that, as $\epsilon \rightarrow 0$, the least energy solution has a spike near the boundary if $\lambda \le \lambda _{*} $, and has an interior spike near the innermost part of the domain if $\lambda &gt; \lambda _{*} $. Central to our study is the corresponding problem on the half space.
LA - eng
UR - http://eudml.org/doc/84496
ER -

References

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  1. [1] P. Bates – G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns. 160 (2000), 283-356. Zbl0990.35016MR1737000
  2. [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. Zbl1157.35407MR1667283
  3. [3] C. C. Chen – C. S. Lin, Uniqueness of the ground state solution of Δ u + f ( u ) = 0 in ℝ N , N ≥ 3 , Comm. PDE. 16 (1991), 1549-1572. Zbl0753.35034MR1132797
  4. [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. Zbl0932.35080MR1736974
  5. [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. Zbl0942.35058MR1742305
  6. [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. Zbl0949.35054MR1737546
  7. [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. Zbl0829.92001MR1279745
  8. [8] M. del Pino – P. Felmer – J. Wei, Mutiple peak solutions for some singular perturbation problems, Cal. Var. PDE 10 (2000), 119-134. Zbl0974.35041MR1750734
  9. [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. Zbl1087.35507MR1710015
  10. [10] E. N. Dancer – S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math. 189 (1999), 241-262. Zbl0933.35070MR1696122
  11. [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. Zbl0506.35035MR688279
  12. [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. Zbl0625.35030MR877892
  13. [13] C. Gui – J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Diff. Eqns. 158 (1999), 1-27. Zbl1061.35502MR1721719
  14. [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. Zbl0949.35052MR1758231
  15. [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. Zbl0944.35020MR1743431
  16. [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. Zbl0469.35052MR634248
  17. [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. Zbl0964.35047MR1782991
  18. [18] M. K. Kwong, Uniqueness of positive solutions of Δ u - u + u p = 0 in R n , Arch. Rational Mech. Anal. 105 (1989), 243-266. Zbl0676.35032MR969899
  19. [19] Y.-Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. PDE 23 (1998), 487-545. Zbl0898.35004MR1620632
  20. [20] Y.-Y. Li – L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), 1445-1490. Zbl0933.35083MR1639159
  21. [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. Zbl0541.49009MR778970
  22. [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. Zbl0704.49004MR778974
  23. [23] P. L. Lions, “Generalized solutions of Hamilton-Jacobi equations”, Pitman, 1982. Zbl0497.35001MR667669
  24. [24] C.-S. Lin – W.-M. Ni – I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns. 72 (1988), 1-27. Zbl0676.35030MR929196
  25. [25] W.-M. Ni, Diffusion, cross-diffusion, and their spike- layer steady states, Notices of Amer. Math. Soc. 45 (1998), 9-18. Zbl0917.35047MR1490535
  26. [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. Zbl0754.35042MR1115095
  27. [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. Zbl0796.35056MR1219814
  28. [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. Zbl0843.35006MR1337211
  29. [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. Zbl0946.35007MR1639546
  30. [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. Zbl0838.35009MR1342381
  31. [31] J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqns. 129 (1996), 315-333. Zbl0865.35011MR1404386
  32. [32] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns. 134 (1997), 104-133. Zbl0873.35007MR1429093
  33. [33] J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. 50 (1998), 159-178. Zbl0918.35024MR1622042
  34. [34] J. Wei, On the effect of the domain geometry in a singularly perturbed Dirichlet problem, Diff. Int. Eqns. 13 (2000), 15-45. Zbl0970.35034MR1811947
  35. [35] J. Wei – M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 459-492. Zbl0910.35049MR1632937
  36. [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. Zbl0922.35025MR1709667

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