Unique continuation for Schrodinger operators with unbounded potentials

We consider unique continuation theorems for solution of inequalities $\text{¦Δu(x)¦ ? W(x) ¦u(x)¦}$ with W allowed to be unbounded. We obtain two kinds of results. One allows W ? L^p_loc($\text{R}$ⁿ) with $\text{p ? n ? 2}\text{for}\text{n > 5, p >}\text{1}\text{3}\text{(2n ? 1)}\text{for}\text{n ? 5}$. The other requires fW² to be ?Δ-form bounded for all f ? C₀^∞.     

Symmetry results for nonlinear elliptic operators with unbounded drift

We prove the one-dimensional symmetry of solutions to elliptic equations of the form ?div(e ^G(x) a(|?u|)?u) = f(u) e ^G(x), under suitable energy conditions. Our results holds without any restriction on the dimension of the ambient space.     

Smoothing estimates for the Schrödinger equation with unbounded potentials

We prove a local in time smoothing estimate for a magnetic Schrödinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudodifferential techniques are required.     

Weighted Sobolev embedding with unbounded and decaying radial potentials

We prove embedding results of weighted W1,p(RN) spaces of radially symmetric functions. The results then are used to obtain ground and bound state solutions of quasilinear equations with unbounded or decaying radial potentials.     

Existence results for partial neutral integro-differential equation with unbounded delay

In this article, we study the existence and regularity of mild solution for a class of partial neutral integro-differential equation with unbounded delay.     

Compactness of Schrödinger semigroups with unbounded below potentials

By using the super Poincaré inequality of a Markov generator L₀ on L²(μ) over a σ-finite measure space (E,F,μ), the Schrödinger semigroup generated by L₀−V for a class of (unbounded below) potentials V is proved to be L²(μ)-compact provided μ(V?N)<∞ for all N>0. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., Rd under the condition that V(x)→∞ as |x|→∞. Concrete examples are provided to illustrate the main result.     

Existence results for partial neutral functional integrodifferential equations with unbounded delay

We prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the Leray-Schauder alternative.     

关于时滞差分方程有界性的Razumikhin型定理

对时滞差分方程建立了新的关于有界性的Raxumikhin型定理,其中可以避免采用不易寻找的辅助函数P,它包含了已知的结果,由它推出的一些结论更易于应用。     

A new non-parametric detector of univariate outliers for distributions with unbounded support

The purpose of this paper is to construct a new non-parametric detector of univariate outliers and to study its asymptotic properties. This detector is based on a Hill’s type statistic. It satisfies a unique asymptotic behavior for a large set of probability distributions with positive unbounded support (for instance: for the absolute value of Gaussian, Gamma, Weibull, Student or regular variations distributions). We have illustrated our results by numerical simulations which show the accuracy of this detector with respect to other usual univariate outlier detectors (Tukey, MAD or Local Outlier Factor detectors). The detection of outliers in a database providing the prices of used cars is also proposed as an application to real-life database.     

Quasilinear Schrödinger equations with unbounded or decaying potentials

We study the existence of nonnegative and nonzero solutions for the following class of quasilinear Schrödinger equations: where V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity. In order to prove our existence result we used minimax techniques in a suitable weighted Orlicz space together with regularity arguments and we need to obtain a symmetric criticality type result.