N-Laplacian equations in ℝN with 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), where a is a continuous function which is coercive, i.e., a(x)→∞ as |x|→∞ and the nonlinearity f behaves like exp(α|u|N/(N−1)) when |u|→∞.