A semilinear system of Schrödinger–Maxwell equations. (May 2020) - Boccardo, Lucio; Orsina, Luigi
Abstract: In this paper we are going to prove existence and regularity results for positive solutions of the following elliptic system: − div ( M ( x ) ∇ u ) + r φ u r − 1 = f + φ r, − div ( M ( x ) ∇ φ ) + r u φ r − 1 = u r . where Ω is a bounded open subset of R N, M is a bounded, uniformly elliptic matrix, r > 1, and f ≥ 0 belongs to some Lebesgue space L m ( Ω ), with m ≥ 1 . We will also prove the relationships of the solutions of the system with saddle points of the integral functional J ( v, ψ ) = 1 2 ∫ Ω M ( x ) ∇ v ⋅ ∇ v − 1 2 ∫ Ω M ( x ) ∇ ψ ⋅ ∇ ψ + ∫ Ω | v | r ψ − ∫ Ω | ψ | r v − ∫ Ω f v