Related papers: Fractional Differential Couples by Sharp Inequalit…

We consider a fractional variant of Maxwell's equations, where the electric and magnetic fields are modeled as two-point fields. To formulate the system, we introduce a fractional curl operator that is compatible with the fractional…

Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^n$ are established. For each $0<s<1$, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren…

In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in $\mathbb{C}^n$, we obtain new uniqueness…

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the…

This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional…

In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in…

In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma…

The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a…

In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation}…

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…

Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…

Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq…

In this paper we investigate fractional differential equations with Hilfer fractional derivative of order $1<\gamma<2$ and type $\delta \in [0,1]$ in a Banach space. We introduce a family of general fractional cosine operator functions of…

This article analysis differential equations which represents damped and fractional oscillators. First, it is shown that prior to using physical quantities in fractional calculus, it is imperative that they are turned dimensionless.…

It is proved that for $\alpha\in (0,1)$, $Q_\alpha(\rn)$, not only as an intermediate space of $W^{1,n}(\rn)$ and $BMO(\rn)$ but also as an affine variant of Sobolev space $\dot{L}^{2}_\alpha(\rn)$ which is sharply imbedded in…

In the present paper, we study the fractional incompressible Stochastic Navier-Stokes equation on $\mathbb{R}^2$, formally defined as \[ \partial_t v = -\tfrac12 (-\Delta)^\theta v - \lambda v \cdot \nabla v + \nabla p - \nabla^{\perp}…

We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We…

The aim of this paper is to obtain the existence of solutions for the following fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*}…