Solutions for a class of fractional Hamiltonian systems with exponential growth

Abstract

In this paper, we establish a weighted Trudinger–Moser type inequality and as the application of this result by using the Galerkin methods and a linking theorem, we prove the existence of weak solutions for the following class of elliptic systems: $\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{1/2}u+V(x)u=g(x,v),& v>0\mathrm{in}\mathbb{R},\\ (-\mathrm{\Delta }{)}^{1/2}v+V(x)v=f(x,u),& u>0\mathrm{in}\mathbb{R},\end{array}$when the potential V is neither bounded away from zero, nor bounded from above. The nonlinear terms $g(x,s)$ and $f(x,s)$ are superlinear at infinity and have subcritical or critical exponential growth.

Keywords:

• Variational methods
• elliptic systems
• Trudinger–Moser inequality
• critical exponents

2010 AMS Subject Classifications:

• Primary: 35J50
• 35J55
• Secondary: 35Q55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research partially supported by CNPq [grant number 306498/2016-2].