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In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$
and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$
. We also prove an existence result of standing waves and prove their orbital stability.
In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system:
\[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \]where $s,t\in (0,1)$
and the mass $m>0.$
By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$
, $\mathbb {R}^{N}_{+}$
and a coercive epigraph domain $\Omega$
in $\mathbb {R}^{N}$
, respectively.
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations
where 
\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}
$0<\alpha <1$, 
$\Omega = \mathbb {R}^{N}$ or 
$\Omega$ is a smooth bounded domain, 
$\Gamma$ is a singular subset of 
$\Omega$ with fractional capacity zero, 
$f(t)$ is locally bounded and positive for 
$t\in [0,\,\infty )$, and 
$f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in 
$t$ for large 
$t$, rather than for every 
$t>0$. Our main result is that the solutions satisfy the estimate
This estimate is new even for 
\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}
$\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.
We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for 
$u(r,t)$
as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.
The main result is that the ellipticity and the Fredholm property of a
$\Psi $DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If
$\lambda \mapsto a_{\lambda }\in S(1,g)$ is a
$\mathcal {C}^N$,
$0\leq N\leq \infty $, map such that each
$a_{\lambda }^w$ is invertible on
$L^2$, then the mapping
$\lambda \mapsto b_{\lambda }\in S(1,g)$, where
$b_{\lambda }^w$ is the inverse of
$a_{\lambda }^w$, is again of class
$\mathcal {C}^N$. Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.
