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August  2015, 35(8): 3771-3797. doi: 10.3934/dcds.2015.35.3771

Concentrating solutions for an anisotropic elliptic problem with large exponent

1. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241

2. 

IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Bât. A, Ile de Saulcy, 57045 Metz Cedex 1, France

Received  September 2014 Revised  October 2014 Published  February 2015

We consider the following anisotropic boundary value problem $$\nabla (a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, \quad u = 0 \ \ \mbox{on} \ \partial\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of anisotropic coefficient $a(x)$ on the existence of concentrating solutions. We show that at a given strict local maximum point of $a(x)$, there exist arbitrarily many concentrating solutions.
Citation: Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771
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show all references

References:
[1]

Proc. Amer. Math. Soc., 132 (2004), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[2]

Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar

[3]

Calc. Var. Partial Differential Equations, 3 (1995), 67-93. doi: 10.1007/BF01190892.  Google Scholar

[4]

Adv. Diff. Equations, 4 (1999), 1-69.  Google Scholar

[5]

Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.  Google Scholar

[6]

Comm. Pure Appl. Math. 55 (2002), 728-771. doi: 10.1002/cpa.3014.  Google Scholar

[7]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.  Google Scholar

[9]

Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142.  Google Scholar

[10]

Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5.  Google Scholar

[11]

Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[12]

J. Diff. Eqns., 227 (2006), 29-68. doi: 10.1016/j.jde.2006.01.023.  Google Scholar

[13]

Proc. Lond. math. Soc., 94 (2007), 497-519. doi: 10.1112/plms/pdl020.  Google Scholar

[14]

J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.  Google Scholar

[15]

Manuscripta Math., 94 (1997), 337-346. doi: 10.1007/BF02677858.  Google Scholar

[16]

Amer. Math. Soc. Transl., 29 (1963), 295-381.  Google Scholar

[17]

Ann. Inst. H. Poincaré Analyse Non Linéaire, 8 (1991), 159-174.  Google Scholar

[18]

Arch. Rat. Mech. Anal., 49 (1973), 241-269.  Google Scholar

[19]

C. R. Math. Acad. Sci. Paris, 348 (2010), 891-896. doi: 10.1016/j.crma.2010.06.024.  Google Scholar

[20]

Comm. Math. Helv., 76 (2001), 506-514. doi: 10.1007/PL00013216.  Google Scholar

[21]

Adv. Nonlineat Stud., 4 (2004), 15-36.  Google Scholar

[22]

Comm. Part. Diff. Equations, 4 (1979), 1263-1297. doi: 10.1080/03605307908820128.  Google Scholar

[23]

Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.  Google Scholar

[24]

Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar

[25]

J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[26]

Nonlinear Anal., 13 (1989), 1241-1249. doi: 10.1016/0362-546X(89)90009-6.  Google Scholar

[27]

Calc. Var. Partial Differential Equations, 28 (2007), 217-247. doi: 10.1007/s00526-006-0044-y.  Google Scholar

[28]

Comp. Rend. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.  Google Scholar

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