arXiv subjects

Yingshu Lü

Publications and source records attributed to Yingshu Lü.

4 recordsLinked to original sources

arXiv2022

Qualitative properties of solutions for system involving fractional Laplacian

In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-Δ)^{s} u (x)= f(u,\,v), \\ (-Δ)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation} in two different types of domains, one is bounded, and the other is unbounded, where $0<s<1$. To investigate the qualitative properties of solutions for fractional equations, the conventional methods are extension method and moving planes method. However, the above methods have technical limits in asymmetric and convex domains and so on. In this work, we employ the direct sliding method for fractional Laplacian to derive the monotonicity of solutions for (1) in $x_n$ variable in different types of domains. Meanwhile, we develop a new iteration method for systems in the proofs which hopefully can be applied to solve other problems.

math.AP↗

arXiv2022

Maximum principles for Laplacian and fractional Laplacian with critical integrability

In this paper, we study maximum principles for Laplacian and fractional Laplacian with critical integrability. We first consider the critical cases for Laplacian with zero order term and first order term. It is well known that for the Laplacian with zero order term $-Δ+c(x)$ in $B_1$, $c(x)\in L^p(B_1)$($B_1\subset \mathbf{R}^n$), the critical case for the maximum principle is $p=\frac{n}{2}$. We show that the critical condition $c(x)\in {L^{\frac{n}{2}}(B_1)}$ is not enough to guarantee the strong maximum principle. For the Laplacian with first order term $-Δ+\vec{b}(x)$($\vec{b}(x)\in L^p(B_1)$), the critical case is $p=n$. In this case, we establish the maximum principle and strong maximum principle for Laplacian with first order term. We also extend some of the maximum principles above to the fractional Laplacian. We replace the classical lower semi-continuous condition on solutions for the fractional Laplacian with some integrability condition. Then we establish a series of maximum principles for fractional Laplacian under some integrability condition on the coefficients. These conditions are weaker than the previous regularity conditions. The weakened conditions on the coefficients and the non-locality of the fractional Laplacian bring in some new difficulties. Some new techniques are developed.

math.FA↗

arXiv2021

Existence of solutions to a generalized self-dual Chern-Simons equation on graphs

Let $ G=(V,E) $ be a connected finite graph and $ Δ$ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \begin{eqnarray}\label{one1} Δ{u}=-λ{e^{F(u)}[e^{F(u)}-1]^2}+4π\sum_{i=1}^{M}{δ_{p_{j}}}, \end{eqnarray} where \begin{equation} F(u)=\left\{\begin{array}{l} \widetilde{F}(u), \ \quad u\leq0, 0, \quad \quad \quad u>0, \end{array} \right. \end{equation} $ \widetilde{F}(u) $ satisfies $ u=1+{\widetilde {F}(u)}-e^{\widetilde {F}(u)} $, $ λ>0 $, $M$ is any fixed positive integer, $ δ_{p_{j}} $ is the Dirac delta mass at the vertex $p_j$, and $p_1$, $p_2$, $\cdots$, $p_M$ are arbitrarily chosen distinct vertices on the graph. We first prove that there is a critical value $λ_c$ such that if $λ\geqλ_c$, then the generalized self-dual Chern-Simons equation has a solution $u_λ$. Applying the existence result, we develop a new method to construct a solution of the equation which is monotonic with respect to $λ$ when $λ\geqλ_c$. Then we establish that there exist at least two solutions of the equation via the variational method for $λ>λ_c$. Furthermore, we give a fine estimate of the monotone solution which can be applied to other related problems.

math.AP↗

arXiv2020

Analyticity of the solutions to degenerate Monge-Ampère equations

This paper is devoted to study the following degenerate Monge-Ampère equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=Λ_q (-u)^q \quad \text{in}\quad Ω,\\ u=0 \quad\text{on}\quad \partialΩ\end{cases} \end{eqnarray} for some positive constant $Λ_q$. Suppose $Ω\subset\subset \mathbb R^n$ is uniformly convex and analytic. Then the solution of the degenerate Monge-Ampère equation is analytic in $\barΩ$ provided $q\in \mathbb Z^+$.

math.AP↗