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.

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi-Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction. A description of a related transonic flow model can be found in G.-Q.G. Chen, M. Feldman (2015), Philos. Trans. Roy. Soc. A 373:20140276 (arXiv:1412.1509.Here, the potential f = f (x, ξ) : Ω × R N ×N → R is assumed to be a quasiconvex function with respect to the second variable; we refer to Morrey [30] for details.Ball [4,5] was interested in potentials given bywhere det ξ denotes the determinant of the N × N matrix ξ. It is also assumed that g and h are nonnegative convex functions satisfying the growth hypotheses g(ξ) ≥ c 1 |ξ| p and lim t→+∞ h(t) = +∞,

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We present a weighted version of the Caffarelli-Kohn-Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.

In this paper, we study the fractional p(⋅, ⋅)-Laplacian, and we introduce the corresponding nonlocal conormal derivative for this operator. We prove the basic properties of the corresponding function space, and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, see Sec. IV, we prove the existence of weak solutions of corresponding p(⋅, ⋅)-Robin boundary problems with sign-changing potentials by applying variational tools.

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

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