Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition

In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a Ambrosetti-Rabinowitz-type condition. The interaction of the nonlinearities with the spectrum of the fractional operator will used to study the existence and multiplicity of solutions. The main technical result proves that a local minimum in $C_s^0(\bar{\mathrm{\Omega }})$ is also a local minimum in $W_0^{s,p}$ for nonlinearities with exponential growth.