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We obtain nontrivial solutions of an    (  N  ,  q  )   ‐Laplacian problem with a critical Trudinger–Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais–Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

( N,q )‐Laplacian problems with critical Trudinger‐Moser nonlinearities