Nonautonomous fractional Hamiltonian system with critical exponential growth

• Published in: Nonlinear differential equations and applications Vol. 26; no. 4; pp. 1 - 25
• Main Authors: do Ó, João Marcos, Giacomoni, Jacques, Mishra, Pawan Kumar
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2019; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Existence theorems; Hamiltonian functions; Mathematics; Mathematics and Statistics; 35R11; Critical growth nonlinearities of Trudinger–Moser type; Elliptic systems involving square root of the Laplacian; Linking theorem; 35A15; Primary 35J50; critical growth nonlinearities of Trudinger-Moser type; linking theorem; elliptic systems involving square root of the Laplacian
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-019-0575-5

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole R ( - Δ ) 1 2 u + u = Q ( x ) g ( v ) in R , ( - Δ ) 1 2 v + v = P ( x ) f ( u ) in R , where ( - Δ ) 1 2 is the square root Laplacian operator. We assume that the nonlinearities f ,  g have critical growth at + ∞ in the sense of Trudinger–Moser inequality and the nonnegative weights P ( x ) and Q ( x ) vanish at + ∞ . Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system.