Sobolev embeddings with 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...

p-Laplacian problems with critical Sobolev exponent

We use variational methods to study the asymptotic behavior of solutions of p-Laplacian problems with nearly subcritical nonlinearity in general, possibly non-smooth, bounded domains.

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

Vector-valued Inequalities on Herz Spaces and Characterizations of Herz–sobolev Spaces with Variable Exponent

The origin of Herz spaces is the study of characterization of functions and multipliers on the classical Hardy spaces ([1, 8]). By virtue of many authors’ works Herz spaces have became one of the remarkable classes of function spaces in harmonic analysis now. One of the important problems on the spaces is boundedness of sublinear operators satisfying proper conditions. Hernández, Li, Lu and Yan...

On a nonlinear eigenvalue problem in Sobolev spaces with variable exponent

Abstract. We consider a class of nonlinear Dirichlet problems involving the p(x)–Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.