Abstract
We obtain nontrivial solutions of a critical (p, q)-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, 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.
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This work was completed while the third-named author was visiting Università di Reggio Calabria, and he is grateful for the kind hospitality of the host university.
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Candito, P., Marano, S.A. & Perera, K. On a class of critical (p, q)-Laplacian problems. Nonlinear Differ. Equ. Appl. 22, 1959–1972 (2015). https://doi.org/10.1007/s00030-015-0353-y
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DOI: https://doi.org/10.1007/s00030-015-0353-y