Existence of solutions for critical (p, q) -Laplacian equations in ℝ^N

In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the (p,q)-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter λ are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter λ is far from 0. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.