Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

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...

Three solutions to a p(x)-Laplacian problem in weighted-variable-exponent Sobolev space

In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567].

Littlewood-Paley Operators on Morrey Spaces with Variable Exponent

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ *-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

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

Minimizers and symmetric minimizers for problems with critical Sobolev exponent

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces D(R ), where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding D(R) →֒ L ∗ (R , Q) when Q is a non-negative, continuous, bounded function. However if Q has certain symmetry properties then all minim...