Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in RN ✩

In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form −div a(x,∇u) + V (x)|u|N−2u = f (x,u) |x|β +εh(x) (0.1) in RN when f : RN ×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. © 2011 Elsevier Inc. All rights reserved.