Clustered solutions around harmonic centers to a coupled elliptic system
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 4, p. 605-628
@article{AIHPC_2007__24_4_605_0,
     author = {D'Aprile, Teresa and Wei, Juncheng},
     title = {Clustered solutions around harmonic centers to a coupled elliptic system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {24},
     number = {4},
     year = {2007},
     pages = {605-628},
     doi = {10.1016/j.anihpc.2006.04.003},
     zbl = {pre05181994},
     mrnumber = {2334995},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2007__24_4_605_0}
}
D'Aprile, Teresa; Wei, Juncheng. Clustered solutions around harmonic centers to a coupled elliptic system. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 4, pp. 605-628. doi : 10.1016/j.anihpc.2006.04.003. http://www.numdam.org/item/AIHPC_2007__24_4_605_0/

[1] Ambrosetti A., Badiale M., Cingolani S., Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal. 140 (1997) 285-300. | MR 1486895 | Zbl 0896.35042

[2] Ambrosetti A., Malchiodi A., Secchi S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Rational Mech. Anal. 159 (3) (2001) 253-271. | MR 1857674 | Zbl 1040.35107

[3] Ambrosetti A., Malchiodi A., Ni W.M., Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, I, Comm. Math. Phys. 235 (3) (2003) 427-466. | MR 1974510 | Zbl 1072.35019

[4] Ambrosetti A., Malchiodi A., Ni W.M., Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, II, Indiana Univ. Math. J. 53 (2) (2004) 297-329. | MR 2056434 | Zbl 1081.35008

[5] Bandle C., Flucher M., Harmonic radius and concentration of energy, hyperbolic radius and Liouville’s equations, Δu=e u and Δu=u n+2 n-2 , SIAM Rev. 38 (2) (1996) 191-238. | MR 1391227 | Zbl 0857.35034

[6] Benci V., Fortunato D., An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (2) (1998) 283-293. | Zbl 0926.35125

[7] Cardaliguet P., Rabah T., On the strict concavity of the harmonic radius in dimension N3, J. Math. Pures Appl. 81 (3) (2002) 223-240. | MR 1894062 | Zbl 1027.31003

[8] Chen X., Del Pino M., Kowalczyk M., The Gierer & Meinhardt system: the breaking of homopclinics and multi-bump ground states, Commun. Contemp. Math. 3 (3) (2001) 419-439. | MR 1849649 | Zbl 1003.34025

[9] Chen C.C., Lin C.S., Uniqueness of the ground state solution of Δu+fu=0 in R N , N3, Comm. Partial Differential Equations 16 (8-9) (1991) 1549-1572. | Zbl 0753.35034

[10] Coclite G.M., Georgiev V., Solitary waves for Maxwell-Schrödinger equations, Electronic J. Differential Equations 2004 (94) (2004) 1-31. | Zbl 1064.35180

[11] Dancer E.N., A note on asymptotic uniqueness for some nonlinearities which change sign, Bull. Austral. Math. Soc. 61 (2000) 305-312. | MR 1748710 | Zbl 0945.35031

[12] Dancer E.N., Wei J., On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 127 (4) (1997) 691-701. | MR 1465415 | Zbl 0882.35052

[13] Dancer E.N., Yan S., Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math. 189 (2) (1999) 241-262. | MR 1696122 | Zbl 0933.35070

[14] D'Aprile T., Mugnai D., Existence of solitary waves for the nonlinear Klein-Gordon Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (5) (2004) 893-906. | Zbl 1064.35182

[15] D'Aprile T., Wei J., On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal. 37 (1) (2005) 321-342. | Zbl 1096.35017

[16] D'Aprile T., Wei J., Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations 25 (1) (2006) 105-137. | Zbl pre05009622

[17] Del Pino M., Felmer P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (2) (1996) 121-137. | MR 1379196 | Zbl 0844.35032

[18] Del Pino M., Felmer P., Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (2) (1998) 127-149. | Numdam | MR 1614646 | Zbl 0901.35023

[19] Del Pino M., Felmer P., Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1) (1997) 245-265. | MR 1471107 | Zbl 0887.35058

[20] Del Pino M., Felmer P., Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann. 324 (1) (2002) 1-32. | MR 1931757 | Zbl 1030.35031

[21] Del Pino M., Kowalczyk M., Wei J., Multi-bump ground states for the Gierer-Meinhardt system in R 2 , Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (1) (2003) 53-85. | Numdam | Zbl 1114.35065

[22] Evans L.C., Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. | MR 1625845 | Zbl 0902.35002

[23] Floer A., Weinstein A., Nonspreading wave pockets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (3) (1986) 397-408. | MR 867665 | Zbl 0613.35076

[24] Gidas B., Ni W.M., Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in R N , Adv. Math. Suppl. Stud. 7A (1981) 369-402. | MR 634248 | Zbl 0469.35052

[25] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2001. | MR 1814364 | Zbl 0361.35003

[26] Grossi M., Some results on a class of nonlinear Schrödinger equations, Math. Z. 235 (4) (2000) 687-705. | MR 1801580 | Zbl 0970.35039

[27] Gui C., Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (3) (1996) 739-769. | MR 1408543 | Zbl 0866.35039

[28] Gui C., Wei J., On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (3) (2000) 522-538. | MR 1758231 | Zbl 0949.35052

[29] Gui C., Wei J., Winter M., Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (1) (2000) 47-82. | Numdam | MR 1743431 | Zbl 0944.35020

[30] Hardin D.P., Saff E.B., Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc. 51 (10) (2004) 1186-1194. | MR 2104914 | Zbl 1095.49031

[31] Helms L.L., Introduction to Potential Theory, John Wiley & Sons Inc., New York, 1969. | MR 261018 | Zbl 0188.17203

[32] Jeanjean L., Tanaka K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations 21 (3) (2004) 287-318. | MR 2094325 | Zbl 1060.35012

[33] Kang X., Wei J., On interacting bumps of semiclassical states of nonlinear Schrödinger equations, Adv. Differential Equations 5 (7-9) (2000) 899-928. | Zbl pre01700753

[34] Kwong M.K., Uniqueness of positive solutions of Δu-u+u p =0 in R N , Arch. Rational Mech. Anal. 105 (3) (1989) 243-266. | MR 969899 | Zbl 0676.35032

[35] Li Y.Y., On a singularly perturbed elliptic equation, Adv. Differential Equations 6 (2) (1997) 955-980. | MR 1606351 | Zbl 1023.35500

[36] Ni W.M., Takagi I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (7) (1991) 819-851. | MR 1115095 | Zbl 0754.35042

[37] Oh Y.J., Existence of semi-classical bound states of nonlinear Schrödinger equation with potential on the class V a , Comm. Partial Differential Equations 13 (12) (1988) 1499-1519. | MR 970154

[38] Pistoia A., Multi-peak solutions for a class of nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (1) (2002) 69-91. | MR 1891696 | Zbl 1001.35030

[39] Rabinowitz P., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (2) (1992) 270-291. | MR 1162728 | Zbl 0763.35087

[40] Ruiz D., Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci. 15 (1) (2005) 141-164. | Zbl 1074.81023

[41] Wang X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (2) (1993) 229-244. | MR 1218300 | Zbl 0795.35118

[42] Wei J., On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (2) (1996) 315-333. | MR 1404386 | Zbl 0865.35011

[43] J. Wei, M. Winter, Symmetric and asymmetric multiple clusters in a reaction-diffusion system, NoDEA Nonlinear Differential Equations Appl., in press. | MR 2374210 | Zbl pre05242837

[44] Wei J., Winter M., Clustered spots in the FitzHugh-Nagumo system, J. Differential Equations 213 (1) (2005) 121-145. | Zbl pre02182110