arXiv subjects

Leyun Wu

Publications and source records attributed to Leyun Wu.

At least 19 recordsLinked to original sources

arXiv2026

Curvature estimate for the heteroclinical solution to a Bose-Einstein condensation system

We investigate heteroclinical solutions of a vector-valued Bose-Einstein condensation system involving the $p$-Laplacian. The main difficulty comes from the degeneracy of the $p$-Laplacian and the possible nonsmooth behavior of the potential wells. Under suitable assumptions on the double-well potential, we establish the existence and detailed asymptotic behavior of heteroclinical solutions. In particular, we prove the strict monotonicity of every component, classify the blow-up profiles near the potential wells through Weiss-type monotonicity formulas, and obtain an almost homogeneity property of the solution. Furthermore, after re-parametrizing the heteroclinical solution by its $l_p$-arc length, we derive a curvature estimate for the trajectory near the potential wells. These results provide the geometric control needed for the construction of radial barrier functions for the corresponding Bose-Einstein condensation system.

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arXiv2026

Pointwise arbitrarily high-order interior estimates for mixed local and nonlocal elliptic equations

In this paper, we mainly focus on pointwise, arbitrarily high-order interior estimates for the mixed local-nonlocal elliptic equation \begin{equation*} (-\Delta)^{s}u(x)-\Delta u(x)=f(x),\quad x\in B_r(0) \end{equation*} with $0<s<1$. The main challenges in this setting are the absence of an explicit Green function and the ineffectiveness of standard bootstrap arguments. Our approach overcomes these difficulties via the Campanato iteration method, which inductively constructs polynomials approximating the solution. Using the fact that all functions are locally $s$-harmonic up to a small error significantly reduces the computational complexity when studying the regularity of the fractional Laplacian acting on these polynomials. Finally, the regularity estimates are obtained from the analysis of a manageable recursive inequality system.

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arXiv2026

Monotonicity for the fractional semi-linear problem in a half space

In this paper, we study semilinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab. Moreover, our analysis relies solely on the local Lipschitz continuity of the nonlinearity $f$, which is weaker than the conditions imposed in earlier works. As a crucial ingredient, we obtained a boundary H\"{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$. The proof introduces a new idea that may be of independent interest. To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, which ensures that the moving plane procedure can be initiated from $x_n = 0$. We then establish two averaging effects for the solutions to fractional equations. These key ingredients guarantee that the planes can be moved continuously all the way to $x_n = \infty$. Previously, narrow region principle can only be applied to a single narrow region. It is for the first time that we establish a multiple narrow region principle that can be applied simultaneously to finitely many narrow regions. Compared with the traditional approaches, methods based on the {\em averaging effect} require substantially weaker regularity assumptions and can even accommodate unbounded solutions. We believe that these new ideas and techniques develop here will serve as powerful tools in studying qualitative properties of solutions to fractional equations.

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arXiv2026

Liouville theorems for mixed local and nonlocal indefinite equations

We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator $ (-\Delta)^s - \Delta$ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions, and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.

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arXiv2026

On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form $(-\Delta)^s-\Delta$, as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical first-order derivative. A major difficulty in this setting stems from the coexistence of operators with different nonlocal structures and incompatible scaling properties, which obstruct the direct use of classical sliding methods. To address this issue, we develop a refined sliding method suited to mixed local-nonlocal operators. As key technical ingredients, we establish new generalized weighted average inequalities, narrow region principles, and maximum principles in bounded and unbounded domains. These tools enable us to derive monotonicity and one-dimensional symmetry results for mixed elliptic equations in bounded domains, half-spaces, and the whole space, and to extend the analysis to parabolic equations with mixed time derivatives. As an application, we resolve the Gibbons' conjecture for a class of mixed fractional equations.

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arXiv2025

Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$

In this series of papers, we investigate coupled systems arising in the study of two-component Bose-Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the present (second) paper of the series, we focus on the nonlocal problem of the form \begin{equation*} \left\{\begin{aligned} (-Δ)^{s}u+u(u^{2}+v^{2}-1)+v(αuv-ω)=0, (-Δ)^{s}v+v(u^{2}+v^{2}-1)+u(αuv-ω)=0, \end{aligned} \right. \end{equation*} which models the stationary states of Rabi-coupled condensates with inter- and intra-species interactions. We prove that for $\frac{1}{2}\le s<1$, any positive entire solution $(u,v)$ in $\mathbb{R}^3$ satisfying the monotonicity condition $\partial_{x_3}u>0>\partial_{x_3}v$ must be one-dimensional. Moreover, when $0<s<\frac{1}{2}$, the same conclusion holds for monotone solutions in $\mathbb{R}^2$. Our work generalizes classical De Giorgi-type theorems to a new class of nonlocal coupled systems and, to the best of our knowledge, presents the first Liouville-type classification of monotone solutions for Rabi-coupled fractional Bose-Einstein condensates, with particular emphasis on fractional Gross-Pitaevskii models.

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arXiv2025

Multiple solutions to the nonlinear Schrödinger equation with a partial confinement

We consider multiple solutions to the nonlinear Schrödinger equation (NLS) with a partial confinement, which is physically relevant to dynamics of the Bose-Einstein condensate. Our study not only verifies the existence of positive ground state solutions and the nonexistence of least energy sign-changing solutions but also sheds light on the symmetry associated with these solutions. A novel finding is the existence of saddle type nodal solutions with their nodal domains intersecting at the origin. Furthermore, we have developed some innovative techniques such as the method of moving planes and the Hopf lemma for nonlinear Schrödinger equations with partial confinement.

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arXiv2025

Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in $\mathbb{R}^{3}$

In this series of papers, we investigate coupled systems arising in the study of two-component Bose--Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the first paper of the series, we focus on the local problem of the form $\Delta u = u(u^2+v^2-1) + v(\alpha uv - \omega)$, $\Delta v = v(u^2+v^2-1) + u(\alpha uv - \omega)$, and prove that positive global solutions in $\mathbb{R}^3$ satisfying $\partial u/\partial x_3 > 0 > \partial v/\partial x_3$ must be one-dimensional.

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arXiv2025

Sharp Boundary Growth Rate Estimate of the Singular Equation $-Δu=u^{-γ}$ in a Critical Cone

For $γ>0$, we study the sharp boundary growth rate estimate of solutions to the Dirichlet problem of the singular Lane-Emden-Fowler equation \begin{equation*} -Δu=u^{-γ} \end{equation*} in a critical $C^{1,1}$ epigraphical cone $Cone_Σ$. We show that the growth rate estimate exhibits fundamentally different behaviors in the following three cases: $1<γ<2$, $γ=2$, and $γ>2$. Moreover, we obtain the sharp growth rate estimate near the origin for $γ>1$. As a consequence, we show that when $Cone_Σ$ is a $C^{1,1}$ epigraphical cone, the additional solvability condition in \cite[Theorem 1.3]{GuLiZh25} is both sufficient and necessary to achieve the growth rate therein, thereby resolving the main open question left in that paper. With the growth rate estimate, we also derive the optimal modulus of continuity for solutions via the interior Schauder estimate. Our approach is to control the values of a solution $U(x)$ in the region $Ω=Cone_Σ\cap B_{1}$ by introducing a sequence of reference points $p_{k}=\frac{16^{1-k}}{2}\vec{e_{n}}$. From the Green function representation of $U(x)$, we derive a discrete integral equation for the sequence $a_{k}=16^{kϕ}U(p_{k})$. Such a computation converts the original PDE problem into a recursion for a discrete integral equation, which can be effectively analyzed using basic ODE methods.

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arXiv2025

Hele-Shaw limit of chemotaxis-Navier-Stokes flows

This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, showing the convergence to a free boundary problem of Hele--Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele--Shaw framework.

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arXiv2025

Refined regularity for nonlocal elliptic equations and applications

In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$ (-Δ)^s u(x) =f(x),\,\, x\in B_1(0). $$ Specifically, we have derived Hölder, Schauder, and Ln-Lipschitz regularity estimates for any nonnegative solution $u,$ provided that only the local $L^\infty$ norm of $u$ is bounded. These estimates stand in sharp contrast to the existing results where the global $L^\infty$ norm of $u$ is required. Our findings indicate that the local values of the solution $u$ and $f$ are sufficient to control the local values of higher order derivatives of $u$. Notably, this makes it possible to establish a priori estimates in unbounded domains by using blowing up and re-scaling argument. As applications, we derive singularity and decay estimates for solutions to some super-linear nonlocal problems in unbounded domains, and in particular, we obtain a priori estimates for a family of fractional Lane-Emden type equations in $\mathbb{R}^n.$ This is achieved by adopting a different method using auxiliary functions, which is applicable to both local and nonlocal problems.

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arXiv2024

Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$

We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in $\mathbb{R}^3$: \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=Δn- \nabla \cdot (χ(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=Δc-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-Δ)^αu-n\nabla ϕ,\\ \nabla \cdot u=0, \end{cases} \end{eqnarray*} which describes the motion of swimming bacteria or bacillus subtilis suspended to water flows. First, we prove some blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin type criterion ($α>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(α>\frac{1}{2})$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $α\geq \frac{5}{4}$. Furthermore, in the scenario of $\frac{3}{4}<α<\frac{5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if the $L^2$-norm of initial data is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations.

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arXiv2023

Global solvability and stability of an alarm-taxis system

This paper is concerned with the global boundedness and stability of classical solutions to an alarm-taxis system describing the burglar alarm hypothesis as an important mechanism of anti-predation behavior when species are threaten by predators. Compared to the existing prey-taxis systems, the alarm-taxis system has more complicated coupling structure and additionally requires the gradient estimate of the primary predator density to attain the global boundedness of solutions. By the sophisticated coupling energy estimates based on the Neumann semigroup smoothing properties, we establish the existence of globally bounded solutions in two dimensions with Neumann boundary conditions and furthermore prove the global stability of co-existence homogeneous steady states under certain conditions on the system parameters.

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arXiv2022

Uniform a priori estimates for $n$-th order Lane-Emden system in $\mathbb{R}^{n}$ with $n\geq3$

In this paper, we establish uniform a priori estimates for positive solutions to the (higher) critical order superlinear Lane-Emden system in bounded domains with Navier boundary conditions in arbitrary dimensions $n\geq3$. First, we prove the monotonicity of solutions for odd order (higher order fractional system) and even order system (integer order system) respectively along the inward normal direction near the boundary by the method of moving planes in local ways. Then we derive uniform a priori estimates by establishing the precise relationships between the maxima of $u$, $v$, $-Δu$ and $-Δv$ through Harnack inequality. Our results extended the uniform a priori estimates for critical order problems in [18, 19] from $n=2$ to higher dimensions $n\geq3$ and in [6, 8] from one single equation to system. With such a priori estimates, one will be able to obtain the existence of solutions via topological degree theory or continuation argument.

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arXiv2021

Nonexistence of solutions for indefinite fractional parabolic equations

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.

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arXiv2021

Liouville theorems for fractional parabolic equations

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and in the whole space $\mathbb{R}^{n-1} \times \mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.

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arXiv2019

Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities

In this paper, we develop a sliding method for the fractional Laplacian. We first obtain the key ingredients needed in the sliding method either in a bounded domain or in the whole space, such as narrow region principles and maximum principles in unbounded domains. Then using semi-linear equations involving the fractional Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. Some new ideas are introduced. Among which, one is to use Poisson integral representation of $s$-subharmonic functions in deriving the maximum principle, the other is to estimate the singular integrals defining the fractional Laplacians along a sequence of approximate maximum points by using a generalized average inequality. We believe that this new inequality will become a useful tool in analyzing fractional equations.

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arXiv2019

A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions

In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$ must be constant.} A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetric about the plane. To this end, we established a {\em maximum principle} for anti-symmetric functions on any half space. We believe that this {\em maximum principle}, as well as the ideas in the proof, will become useful tools in studying a variety of problems involving nonlinear non-local operators.

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