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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 |
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. |
[2] |
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982.
doi: 10.1007/978-1-4612-5734-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381. |
[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. |
[18] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269. |
[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. |
[20] |
L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514.
doi: 10.1007/PL00013216. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982.
doi: 10.1007/978-1-4612-5734-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381. |
[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. |
[18] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269. |
[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. |
[20] |
L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514.
doi: 10.1007/PL00013216. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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