Existence and Regularity of Positive Solutions for Schrödinger-Maxwell System with Singularity

• Published in: Acta applicandae mathematicae 2024-10, Vol.193 (1), p.2, Article 2
• Main Authors: Sbai, Abdelaaziz, El Hadfi, Youssef, El Ouardy, Mounim
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Elliptic functions; Mathematics; Mathematics and Statistics; Maxwell's equations; Partial Differential Equations; Probability Theory and Stochastic Processes; Regularity; Schrodinger equation; Singularity (mathematics)
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-024-00679-6

In this paper we study the existence of positive solutions for the following Schrödinger–Maxwell system of singular elliptic equations 1 { − div ( A ( x ) ∇ u ) + ψ u r − 1 = f ( x ) u θ  in  Ω , − div ( M ( x ) ψ ) = u r  in  Ω , u , ψ > 0  in  Ω , u = ψ = 0  on  ∂ Ω , where Ω is a bounded open set of R N , N > 2 , r > 1 , 0 < θ < 1 and f is nonnegative function belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by proving how the structure of the system gives rise to a regularizing effect on the summability of the solutions.