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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.

##### MSC:
 35J60 Nonlinear elliptic equations 35B25 Singular perturbations (PDE) 47J30 Variational methods (nonlinear operator equations)
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##### References:
 [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