N-Laplacian equations inℝNwith critical growth

Abstract

We study the existence of nontrivial solutions to the following problem:{u∈W1,N(ℝN),u≥0  and−div(|∇u|N−2∇u)+a(x)|u|N−2u=f(x,u)  in  ℝN(N≥2),whereais a continuous function which is coercive, i.e.,a(x)→∞ as |x|→∞and the nonlinearityfbehaves likeexp(α|u|N/(N−1))when|u|→∞.