Multiplicity of solutions to elliptic equations with exponential nonlinearities

The multiplicity of positive solutions to quasi-linear elliptic equations with exponential nonlinearities is obtained through a singular Trudinger-Moser inequality, which is due to Ref.21, the mountain-pass theorem without the Palais-Smale condition and the Ekeland’s variational principle. In particular, for the proof of our main results, we follow the lines of Refs.11,15.