Electron. J. Differential Equations, Vol. 2018 (2018), No. 174, pp. 1-21.

Nontrivial complex solutions for magnetic Schrodinger equations with critical nonlinearities

Sara Barile, Giovany M. Figueiredo

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.

Submitted September 14, 2017. Published October 22, 2018.
Math Subject Classifications: 35B33, 35J20, 35Q55.
Key Words: Magnetic Schrodinger equations; critical nonlinearities; minimization problem; concentration-compactness methods; Pohozaev manifold.

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Sara Barile
Dipartimento di Matematica
Università degli Studi di Bari Aldo Moro
Via E. Orabona 4, 70125 Bari, Italy
email: sara.barile@uniba.it
Giovany M. Figueiredo
Universidade de Brasilia - UNB
Departamento de Matemática
Campus Universitário Darcy Ribeiro
Brasilia - DF, CEP 70.910-900, Brazil
email: giovany@unb.br

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