Nontrivial complex solutions for magnetic Schrodinger equations with critical nonlinearities

Abstract

Using minimization arguments we establish the existence of a complex solution to the magnetic Schrodinger equation $$ - (\nabla + i A(x) )^2 u + u = f(|u|^2) u \quad \text{in }\mathbb{R}^N, $$ where $N \geq 3$, $A:\mathbb{R}^N \to \mathbb{R}^N$ is the magnetic potential and f satisfies some critical growth assumptions. First we obtain bounds from a real Pohozaev manifold. Then relate them to Sobolev imbedding constants and to the least energy level associated with the real equation in absence of the magnetic field (i.e., with A(x)=0). We also apply the Lions Concentration Compactness Principle to the modula of the minimizing sequences involved.

Keywords

• Magnetic Schrodinger equations
• critical nonlinearities
• minimization problem
• concentration-compactness methods
• Pohozaev manifold