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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:when the potential  V  is neither bounded away from zero, nor bounded from above. The nonlinear termsandare superlinear at infinity and have subcritical or critical exponential growth.

Solutions for a class of fractional Hamiltonian systems with exponential growth