On a class of fractional Laplacian problems with variable exponents and indefinite weights

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

“…In recent years, mathematicians began to gradually consider variable exponent Laplace operator ∆ p(x) and ∆ s p(x,•) , see the literature [41][42][43][44][45][46]. It is worth mentioning that Kaufmann et al in [46] extended the variable exponent Sobolev spaces to the fractional case and established the compact embedding theorem of variable exponent Sobolev spaces.…”

“…where Dv = |∇v| p−2 + |∇v| q−2 . Since the system had a wide range of applications in the field of physics and related sciences, this kind of problem has received much attention, we refer to [1,42,49,[52][53][54][55][56][57]. Such as, in the integer order case, the authors in [54] used the constraint minimization to study the subcritical problem with p&q-Laplacian and proved the existence of this problem without the Ambrosetti-Rabinowitz condition.…”

Existence of Solutions for a Class of Fractional Kirchhoff-type Systems in $\mathbb{R}^N$ with Non-standard Growth

“…In the context of non homogeneous materials (such that electrorheological fluids and smart fluids), the use of Lebesgue and Sobolev spaces L p and W s,p seems to be inadequate, which leads to the study of variable exponent Lebesgue and Sobolev spaces L p(x) and W s,p(x,• ) . Moreover, the study of problems which involves the fractional p(x, • )-Laplacian and the corresponding nonlocal elliptic equations constitutes a promising domain of research in which many mathematicians had contributed (see for example [1,7,11,12,14,21]). Furthermore, this type of problems arise in many physical phenomena such as conservation laws, ultra-materials and water waves, optimization, population dynamics, soft thin films, mathematical finance, phases transitions, stratified materials, anomalous diffusion, crystal dislocation, semipermeable membranes, flames propagation, ultra-relativistic limits of quantum mechanics, we refer the reader to [9,24] for details.…”

Three solutions for fractional $p(x,.)$-Laplacian Dirichlet problems with weight

“…)-Laplacian operator). Then, they introduced a suitable functional space to study the problems in which a fractional variable exponent operator is involved; see, e.g., [7,8,9,10,11,12,13,14,15,16,17] and the reference therein.…”