Related papers: Quantitative uniqueness for the power of Laplacian…

In this paper we study the local behavior of a solution to the Lam\'e system when the Lam\'e coefficients $\lambda$ and $\mu$ satisfy that $\mu$ is Lipschitz and $\lambda$ is essentially bounded in dimension $n\ge 2$. One of the main…

Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the…

We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the…

This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing…

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group $\mathbb G$ for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp…

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold,…

We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Second, we…

We rereview the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and…

We consider the uniqueness of the solution of the surface quasi-geostrophic equation with fractional Laplacian. We show that the uniqueness holds in non-homogeneous Besov spaces without any additional assumption which is supposed to…

We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable…

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to $(\partial_t - \Delta)^s u =Vu$ for $s\in [1/2, 1)$ via new Carleman estimates. Our main result Theorem 1.1 and Theorem 1.3 can be thought of as…

In this paper we prove the existence and uniqueness of positive classical solution of the fractional Laplacian with singular nonlinearity in a smooth bounded domain with zero Drichlet boundary conditions. By the method of sub-supersolution,…

In this work, we establish the existence and multiplicity of weak solutions for nonlocal elliptic problems driven by the fractional $\Phi$-Laplacian operator, in the presence of a sign-indefinite nonlinearity. More specifically, we…

We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of…

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…

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C{^1}$ potential $V$. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and…

We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a…

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered…

We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…