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, Delta u)) + (x)vertical bar u vertical bar(N-2)u= f(x, u)/vertical bar x vertical bar(beta) + epsilon h(x) in R-N when f : R-N x R -> R behaves like exp(alpha vertical bar u vertical bar(N/(N-1))) when vertical bar u vertical bar -> infinity co and satisfies the Ambrosetti-Rabinowitz condition. In particular, in the case of N-Laplacian, i.e., a(x,del u) = vertical bar del u vertical bar(N-2)del u, we obtain multiplicity of weak solutions of (0.1). Moreover, we can get the nontriviality of the solution in this case when epsilon = 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. (C) 2011 Elsevier Inc. All rights reserved.