In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form (0.1) − div ( a ( x , ∇ u ) ) + V ( x ) | u | N − 2 u = f ( x , u ) | x | β + ε h ( x ) in R N when f : R N × R → R behaves like exp ( α | u | N / ( N − 1 ) ) when | u | → ∞ and satisfies the Ambrosetti–Rabinowitz condition. In particular, in the case of N-Laplacian, i.e., a ( x , ∇ u ) = | ∇ u | N − 2 ∇ u , we obtain multiplicity of weak solutions of (0.1). Moreover, we can get the nontriviality of the solution in this case when ε = 0 . Finally, we show that the main results remain true if one replaces the Ambrosetti–Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details.