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 . We prove the existence of solutions concentrating along some closed curve on .
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 en dimension 3. On prouve l'existence de solutions qui se concentrent le long de certaines courbes fermées de .
Accepted:
Published online:
Andrea Malchiodi 1
@article{CRMATH_2004__338_10_775_0, author = {Andrea Malchiodi}, title = {Solutions concentrating at curves for some singularly perturbed elliptic problems}, journal = {Comptes Rendus. Math\'ematique}, pages = {775--780}, publisher = {Elsevier}, volume = {338}, number = {10}, year = {2004}, doi = {10.1016/j.crma.2004.03.023}, language = {en}, }
Andrea Malchiodi. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.023/
[1] 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 or on bounded domains with small diffusion, Preprint
[5] Differential Integral Equations, 16 (2003) no. 3, pp. 349-384
[6] J. Funct. Anal., 149 (1997), pp. 245-265
[7] Canad. J. Math., 52 (2000) no. 3, pp. 522-538
[8] Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin, 1976
[9] Comm. Pure Appl. Math., 51 (1998), pp. 1445-1490
[10] 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] 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] Notices Amer. Math. Soc., 45 (1998) no. 1, pp. 9-18
[17] Comm. Pure Appl. Math., 41 (1991), pp. 819-851
[18] Duke Math. J., 70 (1993), pp. 247-281
[19] Trans. Amer. Math. Soc., 354 (2002) no. 8, pp. 3117-3154
Cited by Sources:
Comments - Policy