Solutions for Singular Critical Growth Schrödinger Equations with Magnetic Field

In this paper, we are concerned with the semilinear Schrödinger equation (1.1) −∆Au− V (x)u = |u| ∗−2u , x ∈ R , where −∆A = (−i∇+A)2, u : R → C, N ≥ 3, 2∗ = 2N N−2 denotes the critical Sobolev exponent, A = (A1, A2, ..., AN ) : R N → R is the vector (or magnetic) potential, the coefficient V is the scalar (or electric) potential and may be signchanging. The nonlinear Schrödinger equation arises in different physical theories (e.g., the description of Bose–Estein condensates and nonlinear optics), and has been widely considered in the literature, see [1, 6, 7, 8, 11, 13]. Throughout this paper, suppose A ∈ Lloc(R ,R ). Define L(R , V −dx) := {