On the Schrödinger equation involving a critical Sobolev exponent and magnetic field

Jan Chabrowski, Andrzej Szulkin

DOI: http://dx.doi.org/10.12775/TMNA.2005.001

Abstract


We consider the semilinear Schrödinger equation
$$
-\Delta_A u+V(x)u =Q(x)|u|^{2^{*}-2}u.
$$
Assuming that $V$ changes sign, we establish the
existence of a solution $u\ne 0$ in the Sobolev space $ H_{A,V^+}^{1}(\RN)$.
The solution is obtained by a min-max type argument based on
a topological linking. We also establish certain regularity properties
of solutions for a rather general class of equations involving the
operator $-\Delta_A$.

Keywords


Semilinear Schrödinger equation; critical Sobolev exponent; magnetic field; linking

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