A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces.

On weighted critical imbeddings of Sobolev spaces

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that Lexp tα(χB) = Lexp tα(ρ) if and only if ρq ∈ Lq with some q > 1. Second, we consider the Sobolev space W ...

Dimension-free imbeddings of Sobolev spaces

We prove dimension-free imbedding theorems for Sobolev spaces using extrapolation means and the Gross logarithmic inequality.

On The Sobolev-type Inequality for Lebesgue Spaces with a Variable Exponent

Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Abstract. We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R . We focus on the cases when f±(x, u) = ±(−λ|u| u+ |u|u), where m(x) := max{p1(x), p2(x)} < q(x) < N ·m(x) N−m(x) for any x ∈ Ω. In the first case we show the existence of infinitely many weak solutions for any λ > 0. In the second case we prove that if λ is...

Extrapolation of Reduced Sobolev Imbeddings

We consider fractional Sobolev spaces with dominating mixed derivatives and prove generalizations of Trudinger’s limiting imbedding theorem.