Existence of solutions for critical (p, q)-Laplacian equations in RN

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 lambda 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 lambda is far from 0. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.