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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

Liping Wang 1, and  Dong Ye 2,

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.
Keywords: large exponent, multi-bubble concentration., Anisotropic elliptic problem.
Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J4.
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
References:
[1]

Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar

[3]

A. Bahri, Y. Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, 3 (1995), 67-93. doi: 10.1007/BF01190892.  Google Scholar

[4]

P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Equations, 4 (1999), 1-69.  Google Scholar

[5]

D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.  Google Scholar

[6]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. doi: 10.1002/cpa.3014.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

E. N. Dancer and S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.  Google Scholar

[9]

M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142.  Google Scholar

[10]

M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5.  Google Scholar

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P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[12]

P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Diff. Eqns., 227 (2006), 29-68. doi: 10.1016/j.jde.2006.01.023.  Google Scholar

[13]

P. Esposito, M. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. math. Soc., 94 (2007), 497-519. doi: 10.1112/plms/pdl020.  Google Scholar

[14]

P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbbR^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.  Google Scholar

[15]

M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346. doi: 10.1007/BF02677858.  Google Scholar

[16]

I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381.  Google Scholar

[17]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8 (1991), 159-174.  Google Scholar

[18]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269.  Google Scholar

[19]

S. Khenissy, Y. Rébaï and D. Ye, Expansion of the Green's function for divergence form operators, C. R. Math. Acad. Sci. Paris, 348 (2010), 891-896. doi: 10.1016/j.crma.2010.06.024.  Google Scholar

[20]

L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514. doi: 10.1007/PL00013216.  Google Scholar

[21]

K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlineat Stud., 4 (2004), 15-36.  Google Scholar

[22]

F. Mignot, F. Murat and J. P. Puel, Variation d'un point retourment par rapport au domaine, Comm. Part. Diff. Equations, 4 (1979), 1263-1297. doi: 10.1080/03605307908820128.  Google Scholar

[23]

X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.  Google Scholar

[24]

X. Ren and J. Wei, Singular point condensation and least energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar

[25]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[26]

O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal., 13 (1989), 1241-1249. doi: 10.1016/0362-546X(89)90009-6.  Google Scholar

[27]

J. Wei, D. Ye and F. Zhou, Bubbling slutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247. doi: 10.1007/s00526-006-0044-y.  Google Scholar

[28]

D. Ye, Une remarque sur le comportement asymptotique des solutions de $- \Delta u = \lambda f(u)$, Comp. Rend. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.  Google Scholar

show all references

References:
[1]

Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar

[3]

A. Bahri, Y. Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, 3 (1995), 67-93. doi: 10.1007/BF01190892.  Google Scholar

[4]

P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Equations, 4 (1999), 1-69.  Google Scholar

[5]

D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.  Google Scholar

[6]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. doi: 10.1002/cpa.3014.  Google Scholar

[7]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[8]

E. N. Dancer and S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.  Google Scholar

[9]

M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142.  Google Scholar

[10]

M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5.  Google Scholar

[11]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[12]

P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Diff. Eqns., 227 (2006), 29-68. doi: 10.1016/j.jde.2006.01.023.  Google Scholar

[13]

P. Esposito, M. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. math. Soc., 94 (2007), 497-519. doi: 10.1112/plms/pdl020.  Google Scholar

[14]

P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbbR^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.  Google Scholar

[15]

M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346. doi: 10.1007/BF02677858.  Google Scholar

[16]

I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381.  Google Scholar

[17]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8 (1991), 159-174.  Google Scholar

[18]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269.  Google Scholar

[19]

S. Khenissy, Y. Rébaï and D. Ye, Expansion of the Green's function for divergence form operators, C. R. Math. Acad. Sci. Paris, 348 (2010), 891-896. doi: 10.1016/j.crma.2010.06.024.  Google Scholar

[20]

L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514. doi: 10.1007/PL00013216.  Google Scholar

[21]

K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlineat Stud., 4 (2004), 15-36.  Google Scholar

[22]

F. Mignot, F. Murat and J. P. Puel, Variation d'un point retourment par rapport au domaine, Comm. Part. Diff. Equations, 4 (1979), 1263-1297. doi: 10.1080/03605307908820128.  Google Scholar

[23]

X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.  Google Scholar

[24]

X. Ren and J. Wei, Singular point condensation and least energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.  Google Scholar

[25]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[26]

O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal., 13 (1989), 1241-1249. doi: 10.1016/0362-546X(89)90009-6.  Google Scholar

[27]

J. Wei, D. Ye and F. Zhou, Bubbling slutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247. doi: 10.1007/s00526-006-0044-y.  Google Scholar

[28]

D. Ye, Une remarque sur le comportement asymptotique des solutions de $- \Delta u = \lambda f(u)$, Comp. Rend. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.  Google Scholar

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