Partial Differential Equations
Solutions concentrating at curves for some singularly perturbed elliptic problems
Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 775-780.

We study positive solutions of the equation −ε2Δu+u=up, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain Ω. We prove the existence of solutions concentrating along some closed curve on Ω.

On étudie les solutions positives de l'équation −ε2Δu+u=up, où p>1 et ε>0 est petit, avec conditions de Neumann sur le bord sur un domaine Ω en dimension 3. On prouve l'existence de solutions qui se concentrent le long de certaines courbes fermées de Ω.

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DOI: 10.1016/j.crma.2004.03.023
Malchiodi, Andrea 1

1 SISSA, via Beirut 2-4, 34014 Trieste, Italy
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Malchiodi, Andrea. Solutions concentrating at curves for some singularly perturbed elliptic problems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 775-780. doi : 10.1016/j.crma.2004.03.023. http://www.numdam.org/articles/10.1016/j.crma.2004.03.023/

[1] Ambrosetti, A.; Malchiodi, A.; Ni, W.-M. Commun. Math. Phys., 235 (2003), pp. 427-466

[2] A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II, Indiana Univ. Math. J., in press

[3] S. Cingolani, A. Pistoia, Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent, Z. Angew. Math. Phys., in press

[4] E.N. Dancer, Stable and finite Morse index solutions on n or on bounded domains with small diffusion, Preprint

[5] D'Aprile, T. Differential Integral Equations, 16 (2003) no. 3, pp. 349-384

[6] Del Pino, M.; Felmer, P. J. Funct. Anal., 149 (1997), pp. 245-265

[7] Gui, C.; Wei, J. Canad. J. Math., 52 (2000) no. 3, pp. 522-538

[8] Kato, T. Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin, 1976

[9] Li, Y.Y.; Nirenberg, L. Comm. Pure Appl. Math., 51 (1998), pp. 1445-1490

[10] Lin, C.-S.; Ni, W.-M.; Takagi, I. J. Differential Equations, 72 (1988), pp. 1-27

[11] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Preprint

[12] Malchiodi, A.; Montenegro, M. Comm. Pure Appl. Math., 15 (2002), pp. 1507-1568

[13] A. Malchiodi, M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem, Duke Univ. Math. J., in press

[14] A. Malchiodi, W.-M. Ni, J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Preprint

[15] R. Mazzeo, F. Pacard, Foliations by constant mean curvature tubes, Preprint

[16] Ni, W.-M. Notices Amer. Math. Soc., 45 (1998) no. 1, pp. 9-18

[17] Ni, W.-M.; Takagi, I. Comm. Pure Appl. Math., 41 (1991), pp. 819-851

[18] Ni, W.-M.; Takagi, I. Duke Math. J., 70 (1993), pp. 247-281

[19] Shi, J. Trans. Amer. Math. Soc., 354 (2002) no. 8, pp. 3117-3154

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