On the profile of the changing sign mountain pass solutions for an elliptic problem
HTML articles powered by AMS MathViewer
- by E. N. Dancer and Shusen Yan PDF
- Trans. Amer. Math. Soc. 354 (2002), 3573-3600 Request permission
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
- Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
- Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510
- Daomin Cao, Norman E. Dancer, Ezzat S. Noussair, and Shunsen Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems 2 (1996), no. 2, 221–236. MR 1382508, DOI 10.3934/dcds.1996.2.221
- Daomin Cao and Tassilo Küpper, On the existence of multipeaked solutions to a semilinear Neumann problem, Duke Math. J. 97 (1999), no. 2, 261–300. MR 1682241, DOI 10.1215/S0012-7094-99-09712-0
- Philippe Clément and Guido Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 97–121. MR 937538
- E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. (3) 53 (1986), no. 3, 429–452. MR 868453, DOI 10.1112/plms/s3-53.3.429
- E. N. Dancer, K. Y. Lam, and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations, Abstr. Appl. Anal. 3 (1998), no. 3-4, 293–318. MR 1749413, DOI 10.1155/S1085337598000578
- E. N. Dancer and Juncheng Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 691–701. MR 1465415, DOI 10.1017/S0308210500023775
- E. N. Dancer and Juncheng Wei, On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 157 (1999), no. 1, 82–101. MR 1710015, DOI 10.1006/jdeq.1998.3619
- E. N. Dancer and Shusen Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), no. 2, 241–262. MR 1696122, DOI 10.2140/pjm.1999.189.241
- E. N. Dancer and Shusen Yan, A singularly perturbed elliptic problem in bounded domains with nontrivial topology, Adv. Differential Equations 4 (1999), no. 3, 347–368. MR 1671254
- E. N. Dancer and Shusen Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), no. 4, 1177–1212. MR 1757072, DOI 10.1512/iumj.1999.48.1827
- Manuel Del Pino and Patricio L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), no. 3, 883–898. MR 1736974, DOI 10.1512/iumj.1999.48.1596
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1–27. MR 1721719, DOI 10.1016/S0022-0396(99)80016-3
- Changfeng Gui, Juncheng Wei, and Matthias Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 1, 47–82 (English, with English and French summaries). MR 1743431, DOI 10.1016/S0294-1449(99)00104-3
- E.F.Keller and L.A.Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Bio., 26(1970), 399–415.
- Jaeduck Jang, On spike solutions of singularly perturbed semilinear Dirichlet problem, J. Differential Equations 114 (1994), no. 2, 370–395. MR 1303033, DOI 10.1006/jdeq.1994.1154
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- Yan Yan Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998), no. 3-4, 487–545. MR 1620632, DOI 10.1080/03605309808821354
- H.Meinhardt, Models of biological pattern formation, Academic Press, 1982.
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Wei-Ming Ni, Izumi Takagi, and Juncheng Wei, On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: intermediate solutions, Duke Math. J. 94 (1998), no. 3, 597–618. MR 1639546, DOI 10.1215/S0012-7094-98-09424-8
- Wei-Ming Ni and Juncheng Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), no. 7, 731–768. MR 1342381, DOI 10.1002/cpa.3160480704
- G. Sweers, Some results for a semilinear elliptic problem with a large parameter, Proceedings of the first international conference on industrial and applied mathematics (ICIAM 87) (Paris, 1987) CWI Tract, vol. 36, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1987, pp. 109–116. MR 905694
- Juncheng Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), no. 2, 315–333. MR 1404386, DOI 10.1006/jdeq.1996.0120
- Shusen Yan, On the number of interior multipeak solutions for singularly perturbed Neumann problems, Topol. Methods Nonlinear Anal. 12 (1998), no. 1, 61–78. MR 1677747, DOI 10.12775/TMNA.1998.028
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
- Received by editor(s): December 4, 2001
- Received by editor(s) in revised form: February 19, 2002
- Published electronically: May 8, 2002
- Additional Notes: This work was supported by the ARC
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3573-3600
- MSC (2000): Primary 35J60
- DOI: https://doi.org/10.1090/S0002-9947-02-03026-X
- MathSciNet review: 1911512