相关论文: The unique continuation property for a nonlinear e…
We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain…
Let $u: \Omega \subset \mathbb C^n \to \mathbb C^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p'} = 1$ and $\displaystyle \frac{1}{p} -…
We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case…
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong…
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…
We study two types of unique continuation properties for the higher order Schr\"{o}dinger equation with potential $$ i\partial_tu=(-\Delta_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$…
In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-\Delta )_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = \lambda f(x,u) & \quad \mbox{in }\ \Omega\\[0.3cm] u= 0 &…
In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$,…
A two-player stochastic differential game representation has recently been obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…
Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such…
We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or…
In this paper, we consider the following non-linear equations in unbounded domains $\Omega$ with exterior Dirichlet condition: \begin{equation*}\begin{cases} (-\Delta)_p^s u(x)=f(u(x)), & x\in\Omega,\\ u(x)>0, &x\in\Omega,\\ u(x)\leq0,…
For a finite simple undirected graph $G$, let $\gamma(G)$ denote the size of a smallest dominating set of $G$ and $\mu(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities.…
Zero-sum mean payoff games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair $(u,\lambda)$ solution of $T(u)=\lambda \mathbf{1} + u$ where $T:\mathbb{R}^n…
In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the…
This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a…
In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$,…
Let $\Omega\subset \mathbb{R}^N$, $N=1,2,3$, be an open bounded and connected set with continuous piecewise $\mathrm{C}^{\infty}$ boundary. Here we deal with almost periodic distributions of the form $u(t,x)=\sum_{n=0}^{+\infty} c_n S_n(x)…
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,\,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$ such…