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Existence of Solutions for the Nonlinear Schrödinger Equation with V (∞) = 0

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Contributions to Nonlinear Analysis

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 66))

Abstract

We study the existence of a solution of the problem

$$ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right) > 0, \hfill \\ \end{gathered} \right. $$

under the assumption that

$$ \mathop {\lim }\limits_{\left| x \right| \to \infty } V\left( x \right) = 0 $$

where V > 0 and there is no ground state solution.

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, V. Bonanno 25/b, Pisa, Italy

    Vieri Benci, Carlo R. Grisanti & Anna Maria Micheletti

Authors
  1. Vieri Benci
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  2. Carlo R. Grisanti
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  3. Anna Maria Micheletti
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Editor information

Editors and Affiliations

  1. Laboratoire Jacques-Louis Lions B.C. 187, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex 05, France

    Thierry Cazenave

  2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    David Costa

  3. Instituto de Matemática, UNICAMP - IMECC, Caixa Postal: 6065, 13083-859, Campinas, SP, Brasil

    Orlando Lopes

  4. Departamento de Ingenieria Matemática Facultad de Ciencias Fisicas y Matemáticas, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

    Raúl Manásevich

  5. Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI, 53706-1388, USA

    Paul Rabinowitz

  6. Dipartimento di Matematica, Università degli Studi, Via Saldini 50, 20133, Milano, Italy

    Bernhard Ruf

  7. Departamento de Matemática, PUC Rio, Rua Marquês de São Vicente, 225 EdifÍcio Cardeal Leme, Gávea - Rio de Janeiro, 22453-900, Brasil

    Carlos Tomei

Additional information

Dedicated to Professor Djairo de Figueiredo

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Benci, V., Grisanti, C.R., Micheletti, A.M. (2005). Existence of Solutions for the Nonlinear Schrödinger Equation with V (∞) = 0. In: Cazenave, T., et al. Contributions to Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 66. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7401-2_4

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