Bourgain–Brezis–Mironescu formula for magnetic fractional Sobolev spaces with variable exponents

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

This study establishes the existence of two nontrivial weak solutions for a specific class of Kirchhoff-type equations. Utilizing the Mountain Pass Theorem and the Ekeland variational principle, alongside insights from fractional Sobolev spaces with variable orders and exponents, we provide a concise demonstration of our result.

The article is devoted to the study of second-order quasilinear elliptic equations with variable nonlinearity exponents and a locally integrable right-hand side in the space $\mathbb{R}^n$. The author adapts the concept of a locally renormalized solution for equations with variable growth exponents, generalizing the results of M. F. Bidaut-V\'{e}ron and L. V\'{e}ron obtained for equations with constant exponents. The work establishes conditions on the structure of the quasilinear elliptic operator with variable growth that are sufficient for the correct definition of a locally renormalized solution. The author derives a priori local estimates characterizing the regularity of the solution and, based on these, proves the existence of a locally renormalized solution in the space $\mathbb{R}^n$ without additional restrictions on its growth at infinity. Furthermore, the work demonstrates that for a non-negative right-hand side, the solution is also non-negative almost everywhere. The research employs methods of functional analysis, including the theory of Lebesgue and Sobolev spaces with variable exponents. The proofs are based on compactness and monotonicity techniques, as well as the use of special test functions. The results of the work are significant for the theory of nonlinear elliptic equations and can be applied to further studies of degenerate equations and problems with measure-valued data. The study contributes to the development of analytical methods for equations with variable nonlinearity exponents and expands the applicability of the concept of locally renormalized solutions.

We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional p(z, cdot )-Laplacian problems involving potentials with vanishing behavior at infinity as an application.

In this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii’s genus theory, we show the existence and multiplicity of solutions for a Schrodinger–Kirchhoff type equation involving the fractional $$p\left( .,.\right)$$ -Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch–Wang type compact embedding theorem for fractional Sobolev space with variable exponent.

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

In this paper, we extend the fractional Sobolev spaces with variable exponents $$W^{s,p(x,y)}$$ to include the general fractional case $$W^{s,p(x,y)}_K$$ , where p is a variable exponent, $$s\in (0,1)$$ and K is a suitable kernel. We are concerned with some qualitative properties of the space $$W^{s,p(x,y)}_K$$ (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and a compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal p(x, .)-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type $${\mathcal {L}}^{p(x,.)}_K$$ .

In Avikainen (2009, On irregular functionals of SDEs and the Euler scheme. Finance Stoch., 13, 381–401) the author showed that, for any $p,q \in [1,\infty )$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $ \mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}} $, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz–Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy–Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

In this paper, we consider a class of noncooperative critical nonlocal system with variable exponents of the form: where is the gradient of a C1‐function with respect to the variable . We also assume that , here is the critical Sobolev exponent for variable exponents. With the help of the limit index theory and the concentration‐compactness principles for fractional Sobolev spaces with variable exponents, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity.

In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: where M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that , where is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases.

In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.

In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents where is a smooth and bounded domain, , and α are continuous functions on and is a Carathéodory function with . Under suitable assumption on and , first we study the analogous Hardy–Sobolev–Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional [formula omitted]-Laplacian

The sharp affine L2 Sobolev trace inequality and affine energy in the fractional Sobolev spaces

The Dirichlet problem is considered in arbitrary domains for a class of second-order anisotropic elliptic equations with variable nonlinearity exponents and right-hand sides in . It is proved that an entropy solution exists in anisotropic Sobolev spaces with variable exponent. It is proved that the entropy solution obtained is a renormalized solution of the problem under consideration. Bibliography: 37 titles.