On a class of supercritical N-Laplacian problems

We derive a new existence result for a class of N-Laplacian problems where the classical N-Laplacian is replaced by an operator which admits some coefficients depending on the solution itself. Even if such coefficients make the variational approach more difficult, a suitable supercritical growth for the nonlinear term is allowed. Our proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a proper decomposition of the ambient space. Then, a suitable generalization of the Ambrosetti–Rabinowitz Mountain Pass Theorem allows us to establish the existence of at least one nontrivial bounded solution.