A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractionalp(⋅)-Laplacian

Variable order nonlocal Choquard problem with variable exponents

“…Next we state the Sobolev-type embedding theorem for W s(x,y),p(x,x),p(x,y) (R N × R N ). The proof follows using the similar arguments as in [18], where the authors have considered the case s(x, y) = s, constant.…”

The concentration-compactness principles for W s,p(·,·)(ℝ N ) and application

“…As we mentioned above, there have no results for the critical Sobolev type imbedding for the fractional Sobolev spaces with variable exponents. Although the usual critical Sobolev immersion theorem holds in the fractional order or variable exponents setting, we do not know this assertion even in fractional Sobolev spaces with variable exponents de ned in bounded domain; see [8][9][10][11]. Because of this, our rst aim of the present paper is to obtain a critical imbedding from fractional Sobolev spaces with variable exponents into Lebesgue spaces with variable exponents.…”

“…In this section, we recall the fractional Sobolev spaces with variable exponents that was rst introduced in [11], and was then re ned in [10]. Furthermore, we will obtain a critical Sobolev type imbedding on these spaces.…”

“…Recently, many authors have been studied the fractional Sobolev spaces with variable exponents and the corresponding nonlocal equations with variable exponents (see e.g., [8][9][10][11]). To the authors' best knowledge, though most properties of the classical fractional Sobolev spaces have been extended to the fractional Sobolev spaces with variable exponents, there have no results for the critical Sobolev type imbedding for these spaces.…”

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

“…A somehow critical exponent r(•) satisfying (R 1 ) − (R 2 ) already appeared in paper, 12 which deals with problems driven by (−Δ) s p(•) with constant order s(x,y) ≡ s. Ho and Kim 12 introduce a critical embedding and concentration-compactness principles to face nonlocal problems with variable exponents. Following this direction, we refer to the literature [13][14][15][16][17] for other problems with (−Δ) s p(•) , but on a subcritical setting. Instead, papers 1,18-20 concern a local version of operator 21 investigate a class of double phase energy functionals arising in the theory of transonic flows.…”