Related papers: Landis-type conjecture for the half-Laplacian
In this paper we prove strong unique continuation principle and unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the…
Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
In this paper, we consider the indefinite fractional elliptic problem. A corresponding Liouville-type theorem for the indefinite fractional elliptic equations is established. Furthermore, we obtain a priori bound for solutions in a bounded…
In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schr\"odinger operators. We deduce the strong unique continuation property in the presence of subcritical and…
In this article, we study the order of vanishing and a quantitative form of Landis' conjecture in the plane for solutions to second-order elliptic equations with variable coefficients and singular lower order terms. Precisely, we let $A$ be…
We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We prove unique continuation properties for linear variable coefficient Schr\"odinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any…
This paper focuses on the following class of fractional magnetic Schr\"{o}dinger equations \begin{equation*} (-\Delta)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+\lambda\vert u\vert^{q-2}u, \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where…
Consider a solution $u$ to $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$. If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently large absolute constant, then $u\equiv…
In 1974, Landis and Oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time, then the solution must vanish identically before that time, provided the coefficients of the equation satisfy appropriate…
We are interested in nonlinear fractional Schr\"odinger equations with singular potential of form \begin{equation*} (-\Delta)^su=\frac{\lambda}{|x|^{\alpha}}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$,…
Levinson's theorem for the one-dimensional Schr\"{o}dinger equation with a symmetric potential, which decays at infinity faster than $x^{-2}$, is established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger…
We provide a simple and direct proof of a strong-type unique continuation principle for the fractional $p$-Laplacian $(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear…
We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we…
In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that $V\ge 0$ is a measurable real-valued function satisfying $\|V\|_{L^\infty({\mathbb R}^2)} \le 1$. Let $u$ be a real…
We revisit the following fractional Schr\"{o}dinger equation \begin{align}\label{1a} \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes…
We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic…
We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace…
We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with…
We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=v^q&{\rm in}\,\,\R^N,\\ (-\Delta)^\alpha v=u^p&{\rm in}\,\,\R^N, \end{array} \right.…