Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations $$-\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\,x\in \mathbb{R}^{N},$$ where $1<p<N, 0\leq\mu<\left({(N-p)}/{p}\right)^{p}$ and $Q\in L^{\infty}(\mathbb{R}^{N})$

Nonlinear Liouville problems in a quarter plane

We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093)

Remarks on Nondegeneracy of Ground States for Quasilinear Schr\"odinger Equations

In this paper, we answer affirmatively the problem proposed by A. Selvitella in his paper "Nondegenracy of the ground state for quasilinear Schr\"odinger Equations" (see Calc. Var. Partial Differ. Equ., {\bf 53} (2015), pp 349-364): every ground state of equation \begin{eqnarray*}-\Delta u-u\Delta |u|^2+\omega u-|u|^{p-1}u=0&&\text{in }\mathbb{R}^N\end{eqnarray*} is nondegenerate for $10$ is a given constant and $N\ge1$. We also derive further properties on the linear operator associated to ground states of above equation

Gradient Estimates for Solutions To Quasilinear Elliptic Equations with Critical Sobolev Growth and Hardy Potential

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations $$-\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N},$$ where $1<p<N,0\leq\mu<\left((N-p)/p\right)^{p}$ and $Q\in L^{\infty}(\R^{N})$. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity

Fourier decay bound and differential images of self-similar measures

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $\mathcal{I}=\{\rho x+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $\rho\in (0, 1/m)$, any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erd\H{o}s, which is important in estimating the oscillatory integrals. An application to the existence of normal numbers in fractals is also given.Comment: 9 page

Vibrations of an elastic bar, isospectral deformations, and modified Camassa-Holm equations

Vibrations of an elastic rod are described by a Sturm-Liouville system. We present a general discussion of isospectral (spectrum preserving) deformations of such a system. We interpret one family of such deformations in terms of a two-component modified Camassa-Holm equation (2-mCH) and solve completely its dynamics for the case of discrete measures (multipeakons). We show that the underlying system is Hamiltonian and prove its Liouville integrability. The present paper generalizes our previous work on interlacing multipeakons of the 2-mCH and multipeakons of the 1-mCH. We give a unified approach to both equations, emphasizing certain natural family of interpolation problems germane to the solution of the inverse problem for 2-mCH as well as to this type of a Sturm-Liouville system with singular coefficients.Comment: 34 pages, 2 figures. to appear in Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 2, CRC Press, Boca Raton, F

L∞L^\infty-variational problems associated to measurable Finsler structures

We study $L^\infty$-variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1305.6130 by other author

Liouville integrability of conservative peakons for a modified CH equation

The modified Camassa-Holm equation (also called FORQ) is one of numerous $cousins$ of the Camassa-Holm equation possessing non-smoth solitons ($peakons$) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissapative) the Sobolev $H^1$ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the $H^1$ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere (in [3]).Comment: 12 pages, to appear in J. Nonlinear Math. Phy

On the Capacity of p2p Multipoint Video Conference

In this paper, The structure of video conference is formulated and the peer-assisted distribution scheme is constructed to achieve optimal video delivery rate in each sub-conference. The capacity of conference is proposed to referee the video rate that can be supported in every possible scenario. We have proved that, in case of one user watching only one video, 5/6 is a lower bound of the capacity which is much larger than 1/2, the achievable rate of chained approach in [2]. Almost all proofs in this paper are constructive. They can be applied into real implementation directly with a few modifications

The diffuse gamma-ray flux associated with sub-PeV/PeV neutrinos from starburst galaxies

One attractive scenario for the excess of sub-PeV/PeV neutrinos recently reported by IceCube is that they are produced by cosmic rays in starburst galaxies colliding with the dense interstellar medium. These proton-proton ($pp$) collisions also produce high-energy gamma-rays, which finally contribute to the diffuse high-energy gamma-ray background. We calculate the diffuse gamma-ray flux with a semi-analytic approach and consider that the very high energy gamma-rays will be absorbed in the galaxies and converted into electron-position pairs, which then lose almost all their energy through synchrotron radiation in the strong magnetic fields in the starburst region. Since the synchrotron emission goes into energies below GeV, this synchrotron loss reduces the diffuse high-energy gamma-ray flux by a factor of about two, thus leaving more room for other sources to contribute to the gamma-ray background. For a $E_\nu^{-2}$ neutrino spectrum, we find that the diffuse gamma-ray flux contributes about 20% of the observed diffuse gamma-ray background in the 100 GeV range. However, for a steeper neutrino spectrum, this synchrotron loss effect is less important, since the energy fraction in absorbed gamma-rays becomes lower.Comment: Accepted by ApJ, one figure added, small revisions in text, results and conclusions unchange