Anisotropic variable exponent Sobolev spaces and applications to elliptic equations with critical growth

We establish critical embeddings for anisotropic Sobolev spaces with variable exponents, resolving, under log-Hölder continuity supplemented by finite Sobolev regularity and a quantitative integrability-gap condition, an open problem raised by Fan. We derive a Poincaré-type inequality at the harmonic-mean critical scale, give compactness criteria when the critical exponent is attained on a nontrivial set, and prove a corresponding concentration–compactness principle. These results are then applied to anisotropic Leray–Lions equations with critical growth. Under a global quantitative superlinearity condition, the problem possesses a nontrivial weak solution for a sufficiently large reaction parameter. Under the additional oddness assumptions, for every prescribed integer m , it possesses at least m distinct pairs of nontrivial weak solutions once the parameter exceeds a sufficient threshold depending on m .