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A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

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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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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).

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