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Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials. (English) Zbl 1245.35036
Summary: For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeros of the potentials. The results generalize some recent work of {\it A. Ambrosetti, A. Malchiodi} and {\it W.-M. Ni} [C. R., Math., Acad. Sci. Paris 335, No. 2, 145--150 (2002; Zbl 1072.35068)] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in the paper cited above.

35J60Nonlinear elliptic equations
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)
Full Text: DOI
[1] Ambrosetti, A., Felli, V., Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7 , 117-144 (2005) · Zbl 1064.35175 · doi:10.4171/JEMS/24 · http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=7&iss=1&rank=6
[2] Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on n R . Progr. Math. 240, Birkhäuser (2006) · Zbl 1115.35004
[3] Ambrosetti, A., Malchiodi, A., Ni, W.-M.: Singularly perturbed elliptic equations with sym- metry: existence of solutions concentrating on spheres. I. Comm. Math. Phys. 235 , 427-466 (2003) · Zbl 1072.35019 · doi:10.1007/s00220-003-0811-y
[4] Ambrosetti, A., Malchiodi, A., Ruiz, D.: Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Anal. Math., to appear · Zbl 1142.35082 · doi:10.1007/BF02790279
[5] Ambrosetti, A., Ruiz, D.: Radial solutions concentrating on spheres of NLS with vanishing potentials. Preprint · Zbl 1126.35059 · doi:10.1017/S0308210500004789
[6] Ambrosetti, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with vanishing and decaying potentials. Differential Integral Equations 18 , 1321-1332 (2005) · Zbl 1210.35087
[7] Bartsch, T., Peng, S.: Semiclassical symmetric Schrödinger equations: existence of solutions concentrating simultaneously on several spheres. Preprint · Zbl 1133.35087 · doi:10.1007/s00033-006-5111-x
[8] Byeon, J.: Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains. Comm. Partial Differential Equations 22 , 1731-1769 (1997) · Zbl 0883.35040 · doi:10.1080/03605309708821317
[9] Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 165 , 295-316 (2002) · Zbl 1022.35064 · doi:10.1007/s00205-002-0225-6