In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p(x)−Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration–compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain...

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic − →p (·)-Laplace operator, on a bounded open subset of Rn which has a smooth boundary. The abstract framework required to study this kind of differential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable expon...

Let Ω be a bounded smooth domain in R with N ≥ 2 and 1 < p < ∞. We denote by p∗ the critical exponent for the Sobolev trace immersion given by p∗ = p(N − 1)/(N − p) if p < N and p∗ = ∞ if p ≥ N . For any A ⊂ Ω, which is a smooth open subset, we define the space W 1,p A (Ω) = C ∞ 0 (Ω \A), where the closure is taken in W −norm. By the Sobolev Trace Theorem, there is a compact embedding (1.1) W 1...