Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials

We establish the existence and multiplicity of semiclassical bound states of the following nonlinear Schrodinger equation: {−e2Δu+V(x)u=g(x,u)for x∈RN,u(x)→0as |x|→∞ where V changes sign and g is superlinear with critical or supercritical growth as |u|→∞.

[1]  F. Lin,et al.  Solutions of perturbed Schrödinger equations with critical nonlinearity , 2007 .

[2]  Zhi-Qiang Wang,et al.  Standing waves with a critical frequency for nonlinear Schrödinger equations, II , 2003 .

[3]  Y. Oh.  On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential , 1990 .

[4]  Alan Weinstein,et al.  Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential , 1986 .

[5]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[6]  B. Sirakov.  Standing wave solutions of the nonlinear Schrödinger equation in ℝN , 2002 .

[7]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[8]  P. Felmer,et al.  Semi-classical states of nonlinear Schrödinger equations: a variational reduction method , 2002 .

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  Y. Oh.  Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)a , 1988 .

[11]  W. Ni,et al.  On the existence of positive entire solutions of a semilinear elliptic equation , 1986 .

[12]  M. Grossi.  Some results on a class of nonliner Schrödinger equations , 2000 .

[13]  Changfeng Gui,et al.  Existence of multi-bumb solutions for nonlinear schrödinger equations via variational method , 1996 .

[14]  Yanyan Li.  On a singularly perturbed elliptic equation , 1997, Advances in Differential Equations.

[15]  Paul H. Rabinowitz,et al.  Homoclinic type solutions for a semilinear elliptic PDE on ℝn , 1992 .

[16]  A. Pistoia.  Multi-peak solutions for a class of nonlinear Schrödinger equations , 2002 .

[17]  V. Benci.  On critical point theory for indefinite functionals in the presence of symmetries , 1982 .

[18]  M. Pino,et al.  Multi-peak bound states for nonlinear Schrödinger equations , 1998 .

[19]  Jaeyoung Byeon,et al.  Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations , 2002 .

[20]  Xuefeng Wang.  On concentration of positive bound states of nonlinear Schrödinger equations , 1993 .

[21]  Yanheng Ding,et al.  Bound states for semilinear Schrödinger equations with sign-changing potential , 2007 .

[22]  L. Jeanjean,et al.  Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities , 2004 .

[23]  A. Ambrosetti,et al.  Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part II , 2004 .

[24]  A. Ambrosetti,et al.  Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I , 2003 .

[25]  Juncheng Wei,et al.  Nonlinear Schrödinger equations : concentration on weighted geodesics in the semi-classical limit , 2005 .

[26]  A. Ambrosetti,et al.  Semiclassical States of Nonlinear Schrödinger Equations , 1997 .

[27]  Yanheng Ding,et al.  Existence and Number of Solutions for a Class of Semilinear Schrödinger Equations , 2005 .

[28]  Juncheng Wei,et al.  On interacting bumps of semi-classical states of nonlinear Schrödinger equations , 2000, Advances in Differential Equations.

[29]  Juncheng Wei,et al.  Concentration on curves for nonlinear Schrödinger Equations , 2007 .