Fractional Orlicz Sobolev Spaces Research Articles

Ground State Solutions for a Non-Local Type Problem in Fractional Orlicz Sobolev Spaces

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (−ΔΦ)su+V(x)a(|u|)u=f(x,u), x∈RN, where (−ΔΦ)s(s∈(0,1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ-Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V(x). The proof is based on a variant version of the mountain pass theorem and a Lions’ type result in fractional Orlicz–Sobolev spaces.

Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization

AbstractEmbeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

On a class of fractional Γ(.)-Kirchhoff-Schrödinger system type

This paper focuses on the investigation of a Kirchhoff-Schrödinger type elliptic system involving a fractional \(\gamma(.)\)-Laplacian operator. The primary objective is to establish the existence of weak solutions for this system within the framework of fractional Orlicz-Sobolev Spaces. To achieve this, we employ the critical point approach and direct variational principle, which allow us to demonstrate the existence of such solutions. The utilization of fractional Orlicz-Sobolev spaces is essential for handling the nonlinearity of the problem, making it a powerful tool for the analysis. The results presented herein contribute to a deeper understanding of the behavior of this type of elliptic system and provide a foundation for further research in related areas.

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems

Multiple Solutions for a Nonlocal Kirchhoff Problem in Fractional Orlicz-Sobolev Spaces

In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem driven by a nonlocal operator of the elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.

Sign-changing solution for a generalized Kirchhoff problem in the fractional Orlicz-Sobolev space with nonsmooth nonlinearity

In this paper, we study a nonlocal generalized Kirchhoff problem driven by the fractional Orlicz g-Laplace operator and involving a nonsmooth nonlinearity. Although this problem has a double nonlocality, we prove the existence of a least-energy nodal (or sign-changing) solution. Note that the tools used are minimization argument and quantitative deformation lemma.

Multiplicity of solutions for nonlocal parametric elliptic systems in fractional Orlicz–Sobolev spaces

Multiplicity of solutions for nonlocal parametric elliptic systems in fractional Orlicz–Sobolev spaces

Existence and multiplicity of solutions for a Dirichlet problem in fractional Orlicz–Sobolev spaces

Existence and multiplicity of solutions for a Dirichlet problem in fractional Orlicz–Sobolev spaces

A new class of fractional Orlicz-Sobolev space and singular elliptic problems

This paper seeks to introduce a new class of fractional Orlicz-Sobolev space with variable-order. We give basic properties of this space and prove some compactness results. Then, using some techniques of calculus of variations combined with theory of Musielak functions, we prove the existence of a nonnegative weak solution for a singular elliptic type problem in a fractional variable-order Orlicz-Sobolev space with homogeneous Dirichlet boundary conditions.

Existence and multiplicity of solutions to magnetic Kirchhoff equations in Orlicz-Sobolev spaces

In this paper, we study the existence and multiplicity of weak solutions to a general type of Kirchhoff equations in magnetic fractional Orlicz-Sobolev spaces. Specifically, we appeal to Critical Point Theory to prove the existence of non-trivial solutions under the so-called Ambrosetti-Rabinowitz condition. We also state the existence of ground-state solutions. Moreover, multiplicity results which yield the existence of an unbounded sequence of solutions are also provided. Finally, we show existence under a weak-type Ambrosetti-Rabinowitz condition formulated in the framework of Orlicz spaces.

A class of non-local elliptic system in non-reflexive fractional Orlicz–Sobolev spaces

The purpose of this paper is to investigate the existence of a weak solution to a non-local elliptic system, driven by the fractional [Formula: see text]-Laplacian operator in fractional Orlicz–Sobolev space that may be non-reflexive. The non-reflexive case occurs when the Orlicz function [Formula: see text] does not verify the [Formula: see text]-condition.

Boundedness of functions in fractional Orlicz–Sobolev spaces

A necessary and sufficient condition for fractional Orlicz–Sobolev spaces to be continuously embedded into L∞(Rn) is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into L∞(Rn) fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.

Least-energy nodal solutions of nonlinear equations with fractional Orlicz–Sobolev spaces

In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by ( − △ g ) α u + g ( u ) = K ( x ) f ( u ) , in R N , where N ⩾ 3 , ( − △ g ) α is the fractional Orlicz g-Laplace operator, while f ∈ C 1 ( R ) and K is a positive and continuous function. Under a suitable conditions on f and K, we prove a compact embeddings result for weighted fractional Orlicz–Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution.

A Weak Solution to a Non-Local Problem in Fractional Orlicz-Sobolev Spaces

A Weak Solution to a Non-Local Problem in Fractional Orlicz-Sobolev Spaces

Problems involving the fractional g-Laplacian with lack of compactness

In this paper, we prove compact embedding of a subspace of the fractional Orlicz–Sobolev space Ws,GRN consisting of radial functions; our target embedding spaces are of Orlicz type. In addition, we prove a Lions and Lieb type results for Ws,GRN that works together in a particular way to get a sequence whose weak limit is non-trivial. As an application, we study the existence of solutions to quasilinear elliptic problems in the whole space RN involving the fractional g-Laplacian operator, where the conjugated function G̃ of G does not satisfy the Δ2-condition.

A class of elliptic inclusion in fractional Orlicz–Sobolev spaces

ABSTRACT We consider an elliptic inclusion that is driven by ( − Δ ) a ( ⋅ ) s called fractional a ( ⋅ ) -Laplacian operator, with Dirichlet-type boundary conditions. We take two different assumptions on the Carathéodory function f . One of them is where f satisfies the localy Lipchitz condition and the other one is when f does not satisfy an assumption of growth. With a regular Clarke subdifferential ∂ C and Lebourg's mean value theorem, by applying a variational approach together with the critical point theory, we obtain a weak solution to the inclusion problem in appropriate fractional Orlicz–Sobolev spaces.

Interior and up to the boundary regularity for the fractional [formula omitted]-Laplacian: The convex case

We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional g−Laplacian with bounded right hand side and g convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz–Sobolev spaces.

Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

We employing a minimization arguments on appropriate Nehari manifolds, we obtain ground state solutionsfor a non-local elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundaryconditions type.

Ground State Solutions for a Nonlocal System in Fractional Orlicz-Sobolev Spaces

We consider an elliptic system driven by the fractional a . -Laplacian operator, with Dirichlet boundary conditions type. By using the Nehari manifold approach, we get a nontrivial ground state solution on fractional Orlicz–Sobolev spaces.

Maz’ya–Shaposhnikova formula in magnetic fractional Orlicz–Sobolev spaces

In this note we prove the validity of the Maz’ya–Shaposhnikova formula in magnetic fractional Orlicz–Sobolev spaces. This complements a previous asymptotic study of the limit as s ↑ 1 performed by the second author in (Studia Mathematica (2020)).