Nonautonomous fractional Hamiltonian system with critical exponential growth

  • João Marcos do Ó
  • Jacques GiacomoniEmail author
  • Pawan Kumar Mishra


In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole \({\mathbb {R}}\)
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\frac{1}{2}~ u +u=Q(x) g(v)&{}\quad \text{ in } {\mathbb {R}},\\ (-\Delta )^\frac{1}{2}~ v+v = P(x)f(u)&{}\quad \text{ in } {\mathbb {R}}, \end{array}\right. \end{aligned}$$
where \((-\Delta )^\frac{1}{2}\) is the square root Laplacian operator. We assume that the nonlinearities fg have critical growth at \(+\,\infty \) in the sense of Trudinger–Moser inequality and the nonnegative weights P(x) and Q(x) vanish at \(+\infty \). Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system.


Elliptic systems involving square root of the Laplacian Critical growth nonlinearities of Trudinger–Moser type Linking theorem 

Mathematics Subject Classification

Primary 35J50 35R11 35A15 



Research was supported in part by INCTmat/MCT/Brazil, CNPq and CAPES/Brazil.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • João Marcos do Ó
    • 1
  • Jacques Giacomoni
    • 2
    Email author
  • Pawan Kumar Mishra
    • 3
  1. 1.Department of MathematicsBrasília UniversityBrasíliaBrazil
  2. 2.LMAP (UMR E2S-UPPA CNRS 5142), Bat. IPRAPauFrance
  3. 3.Department of MathematicsFederal University of ParaíbaJoão PessoaBrazil

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