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On the profile of the changing sign mountain pass solutions for an elliptic problem

Author(s): E. N. Dancer; Shusen Yan
Journal: Trans. Amer. Math. Soc. 354 (2002), 3573-3600.
MSC (2000): Primary 35J60
Posted: May 8, 2002
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Abstract | References | Similar articles | Additional information

Abstract: We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.


References:

1.
A.Ambrosetti and P.Rabinowitz Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(1973), 349-381. MR 51:6412

2.
H.Berestycki, L.Caffarelli and L.Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa C1. Sci., 25, 69-94(1997). MR 2000e:35053

3.
D.Cao, E.N.Dancer, E.Noussair and S.Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete and Continuous Dynamical Systems, 2(1996), 221-236.MR 96m:35095

4.
D.Cao and T.Küpper, On the existence of the multipeaked solutions to a semilinear Neumman problem, Duke Math. J. 97(1999), 261-300. MR 2000a:35064

5.
P.Clément and G.Sweers, Existence and multiplicity results for a semilinear eigenvalue problem, Ann. Scuola Norm. Sup. Pisa, 14(1987), 97-121. MR 89j:35053

6.
E.N.Dancer, On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large, Proc. London Math. Soc., 53(1986), 429-452. MR 88c:35061

7.
E.N. Dancer, K.Y.Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equation, Abstract and Appl. Anal., 3(1998), 293-318. MR 2001g:35231

8.
E.N.Dancer and J.Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Royal Soc. Edinburgh, 127A(1997), 691-701. MR 98i:35012

9.
E.N.Dancer and J.Wei, On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Diff. Equations, 157(1999), 82-101. MR 2000j:35017

10.
E.N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189(1999), 241-262. MR 2000d:35010

11.
E.N.Dancer and S.Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Diff. Equations, 4(1999), 347-368. MR 2000d:35009

12.
E.N. Dancer and S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana University Math. J., 48(1999), 1177-1212. MR 2001f:35146

13.
M. Del Pino and P.Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math.J., 48(1999), 883-898. MR 2001b:35027
14.
B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68(1979), 209-243.MR 80h:35043

15.
D.Gilbarg and N.S.Trudinger, Elliptic partial differential equations of second order, second edition, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 86c:35035

16.
C. Gui and J.Wei, Multiple interior peak solutions for some singularly perturbed Neumann problem, J. Diff. Equations 158(1999), 1-27. MR 2000g:35035

17.
C. Gui, J.Wei and M.Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problem, Ann. Inst. H. Poincaré Anal., Non Linéaire, 17(2000), 47-82. MR 2001a:35018

18.
E.F.Keller and L.A.Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Bio., 26(1970), 399-415.

19.
J.Jang, On the spike solutions of singularly perturbed semilinear Dirichlet problem, J.Diff. Equations, 114(1994), 370-395. MR 95i:35099

20.
P.L.Lions, The concentration compactness principle in the calculus of variations, the locally compact case, I,II, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1(1984), 109-145, 223-283. MR 87e:49035a; MR 87e:49035b

21.
Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. PDE, 23(1998), 487-545. MR 2000a:35013

22.
H.Meinhardt, Models of biological pattern formation, Academic Press, 1982.

23.
W.M. Ni and I. Takagi, On the shape of the least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41(1991), 819-851. MR 92i:35052

24.
W.M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J.70(1993), 247-281. MR 94h:35072

25.
W.M. Ni, I. Takagi and J.Wei, On the locations and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution, Duke Math. J., 94(1998), 597-618. MR 99h:35011

26.
W.M.Ni and 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 96g:35077

27.
G.Sweers, Some results for a semilinear elliptic problem with a large parameter, Proceedings of the First International Conference on Industrial and Applied Mathematics, Paris-La Villette, 1987, pp. 109-116. MR 89d:35065

28.
J.Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Equations, 129(1996), 315-333. MR 97f:35015

29.
S. Yan, On the number of the interior multipeak solutions for singularly perturbed Neumann problems, Topological Methods in Nonlinear Anal., 12(1999), 61-78. MR 2001c:35024

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Additional Information:

E. N. Dancer
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: normd@maths.usyd.edu.au

Shusen Yan
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: shusen@maths.usyd.edu.au

DOI: 10.1090/S0002-9947-02-03026-X
PII: S 0002-9947(02)03026-X
Received by editor(s): December 4, 2001
Received by editor(s) in revised form: February 19, 2002
Posted: May 8, 2002
Additional Notes: This work was supported by the ARC
Copyright of article: Copyright 2002, American Mathematical Society


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