On the interior spike layer solutions to a singularly perturbed Neumann problem. (English) Zbl 0918.35024

Author’s summary: We construct interior spike layer solutions for a class of semilinear elliptic Neumann problems which arise as stationary solutions of Keller-Segel model in chemotaxis and also as limiting equations for the Gierer-Meinhardt system in biological pattern formation. We also classify the location of single interior peaks. We show exactly how the geometry of the domain affects the spike solution.
Reviewer: I.Ginchev (Varna)


35B40 Asymptotic behavior of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
Full Text: DOI


[1] ADIMURTHI, G. MANCINNI AND S. L. YADAVA, The role of mean curvature in a semilinear Neumann problem involving the critical Sobolev exponent, Comm. Partial Differential Equations 20 (1995), 591-631. · Zbl 0847.35047 · doi:10.1080/03605309508821110
[2] ADIMURTHI, F. PACELLA AND S. L. YADAVA, Interaction between the geometry of the boundary an positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal. 113 (1993), 318-350. · Zbl 0793.35033 · doi:10.1006/jfan.1993.1053
[3] ADIMURTHI, F. PACELLA AND S. L. YADAVA, Characterization of concentration points and L^-estimate for solutions involving the critical Sobolev exponent, Differential Integral Equations 8 (1995), 41-68. · Zbl 0814.35029
[4] E. DANCER AND E. NOUSSAIR, private communication
[5] B. GIDAS, W. -M. Ni AND L. NIRENBERG, Symmetry of positive solutions of nonlinear elliptic equation in R”, Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A (1981), Academic Press, New York, 369-402. · Zbl 0469.35052
[6] M. K. KWONG, Uniqueness of positive solutions of w -u+up =Q in R”, Arch. Rational Mech. Anal 105 (1989), 243-266. · Zbl 0676.35032 · doi:10.1007/BF00251502
[7] C. -S. LIN, W. -M. Ni AND I. TAKAGI, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations 72 (1988), 1-27. · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[8] W. -M. Ni, X. PAN AND I. TAKAGI, Singular behavior of least-energy solutions of a semilinear Neuman problem involving critical Sobolev exponents, Duke Math. J. 67 (1992), 1-20. · Zbl 0785.35041 · doi:10.1215/S0012-7094-92-06701-9
[9] W. -M. Ni AND I. TAKAGI, On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991), 819-851. · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[10] W. -M. Ni AND I. TAKAGI, Locating the peaks of least energy solutions to a semilinear Neuman problem, Duke Math. J. 70 (1993), 247-281. · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4
[11] W. -M. Ni AND I. TAKAGI, Point-condensation generated be a reaction-diffusion system in axiall symmetric domains, Japan J. Industrial Appl. Math. 12 (1995), 327-365. · Zbl 0843.35006 · doi:10.1007/BF03167294
[12] W. -M. Ni AND J. WEI, On the location and profile of spike-layer solutions to singularly perturbe semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731-768. · Zbl 0838.35009 · doi:10.1002/cpa.3160480704
[13] X. B. PAN, Condensation of least-energy solutions of a semilinear Neumann problem, J. Partia Differential Equations 8 (1995), 1-36. · Zbl 0814.35039
[14] X. B. PAN, Condensation of least-energy solutions: the effect of boundary conditions, Nonlinea Analysis: TMA 24 (1995), 195-222. · Zbl 0826.35037 · doi:10.1016/0362-546X(94)E0056-M
[15] X. B. PAN, Further study on the effect of boundary conditions, J. Differential Equations 117 (1995), 446-468. · Zbl 0832.35050 · doi:10.1006/jdeq.1995.1061
[16] X. B. PAN AND X. F. WANG, Semilinear Neumann problem in an exterior domain, Nonlinear Analysis TMA,
[17] X. B. PAN AND XING-WANG Xu, Least energy solutions of semilinear Neumann problems an asymptotics, J. Math. Anal. Appl. 201 (1996), 532-554. · Zbl 0861.35028 · doi:10.1006/jmaa.1996.0272
[18] Z. -Q. WANG, On the existence of multiple single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal. 120 (1992), 375-399. 178J. WEI · Zbl 0784.35035 · doi:10.1007/BF00380322
[19] J. WEI, On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichle problem, J. Differential Equations 129 (1996), 315-333. · Zbl 0865.35011 · doi:10.1006/jdeq.1996.0120
[20] J. WEI, On the effect of the geometry of the domain in a singularly perturbed Dirichlet problem, Differential Integral Equations, · Zbl 1038.94566
[21] J. WEI, On the boundary spike layer solutions of a singularlyperturbed Neumann problem, J. Differentia Equations 134 (1997), 104-133. · Zbl 0873.35007 · doi:10.1006/jdeq.1996.3218
[22] J. WEI, On the interior spike layer solutions for some singular perturbation problems, Proc. Roya Soc. Edinburgh, Section A (Mathematics), · Zbl 0944.35021 · doi:10.1017/S030821050002182X
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