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A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

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Abstract.

We consider the semilinear stationary Schrödinger equation in a magnetic field: (−i∇+A)2 u+V(x)u=g(x,|u|)u in ℝN, where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|2 * −2 and |g(x,|u|)|≤c(1+|u|p −2), where 2<p<2*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.

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Authors and Affiliations

  1. Dip. di Matematica del Politecnico, via Bonardi 9, 20133, Milano, Italy

    Gianni Arioli

  2. Dept. of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Andrzej Szulkin

Authors
  1. Gianni Arioli
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  2. Andrzej Szulkin
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Correspondence to Andrzej Szulkin.

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Communicated by C.A. Stuart

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Arioli, G., Szulkin, A. A Semilinear Schrödinger Equation in the Presence of a Magnetic Field. Arch. Rational Mech. Anal. 170, 277–295 (2003). https://doi.org/10.1007/s00205-003-0274-5

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  • DOI: https://doi.org/10.1007/s00205-003-0274-5

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