On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth

• Published in: Calculus of variations and partial differential equations Vol. 60; no. 1
• Main Authors: Albuquerque, F. S., Carvalho, J. L., Figueiredo, G. M., Medeiros, E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2021; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35J25; 35J60; 35J15
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-020-01902-6

In this paper, we investigate the existence of solutions to the planar non-autonomous Schrödinger–Poisson system - Δ u + V ( | x | ) u + γ ϕ K ( | x | ) u = λ Q ( | x | ) f ( u ) , & x ∈ R 2 , Δ ϕ = K ( | x | ) u 2 , & x ∈ R 2 , where γ , λ are positive parameters, V ,  K ,  Q are continuous potentials, which can be unbounded or vanishing at infinity. By assuming that the nonlinearity f ( s ) has exponential critical growth, we derive the existence of a ground state solution to the system. A key feature of our approach is a new weighted Trudinger–Moser type inequality proved here.