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

#### 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 $-\Delta_A$.

$$

-\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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