We prove existence results in all of $${\mathbb {R}}^N$$ R N for an elliptic problem of (p, q)-Laplacian type involving a critical term, nonnegative weights and a positive parameter $$\lambda $$ λ . In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when $$\lambda $$ λ belongs to a certain interval. Our proofs use variational methods and the concentration compactness principle. Towards this aim we give a detailed proof of tight convergence of a suitable sequence.

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