On the Schrödinger equation involving a critical Sobolev exponent and magnetic field - Chabrowski, Jan; Szulkin, Andrzej (Topological Methods in Nonlinear Analysis, 2005)
We consider the semilinear Schrödinger equation $$-Δ_A u+V(x)u =Q(x)|u|^{2^{*}-2}u. $$Assuming that $V$ changes sign, we establish theexistence of a solution $u e 0$ 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 propertiesof solutions for a rather general class of equations involving theoperator $-Δ_A$.