Omar Benslimane, Ahmed Aberqi, Jaouad Bennouna
The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.
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We consider different fractional Neumann Laplacians of order s, 0<s<1, namely, the Restricted Neumann Laplacian, the Semirestricted Neumann Laplacian and the Spectral Neumann Laplacian. In particular, we are interested in attainability of Sobolev constants for these operators in half-spaces.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.
We prove existence, multiplicity, and bifurcation results for $p$-Laplacian problems involving critical Hardy-Sobolev exponents. Our results are mainly for the case $λ\ge λ_1$ and extend results in the literature for $0 < λ< λ_1$. In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index.
We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces $W^{1,p}$.
We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.
We consider the Benjamin-Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a consequence, unconditional uniqueness for the Benjamin-Ono equation is proved.
Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are precisely the products of the eigenfunctions on the individual factors. There is a related result when Steklov boundary conditions are specified on the second factor. These results enable the characterization of certain Hilbert-Sobolev spaces as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trac…
Existence and uniqueness of solutions in fractional Orlicz-Sobolev spaces for a nonlocal singular elliptic problem
本文研究了一类带边界条件的奇异非线性椭圆问题，证明了其正解的存在性与唯一性，且该结果在分数阶 Orlicz-Sobolev 空间中成立。
This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.
We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two positive constants $λ_{0}$ and $λ_{1}$ with $λ_{0}\leqλ_{1}$ such that $λ_1$ is an eigenvalue of the problem while any value $λ<λ_{0}$ cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of $λ_{1}$ on the right side. We emphasize that throughout the paper no $Δ_2$-condition is needed.
Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian
本文建立一族尖锐Sobolev迹不等式，将Laplacian分数次幂实现为广义Dirichlet-to-Neumann算子，推广了Caffarelli--Silvestre和Yang的结果。
Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings
本文针对特殊平面区域和权重估计加权Poincaré常数，进而证明带密度Neumann拉普拉斯算子第一非零特征值的下界，并指出该特征值可通过改变质量密度任意增大。
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of Hörmander vector fields
Hopf's lemmas and boundary behaviour of solutions to the fractional Laplacian in Orlicz-Sobolev spaces
本文在Orlicz-Sobolev空间框架下，将Hopf边界引理推广到分数阶$a-$拉普拉斯算子，研究满足内部球条件的区域边界附近解的行为，并同时处理常号与变号位势问题。
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
On the uniqueness of eigenfunctions for the vectorial $p$-Laplacian
该研究针对向量值特征函数的非线性特征值问题，给出了对应瑞利商极小元的简洁唯一性证明，推进了向量值p-Laplacian相关理论研究。
On uniqueness theorems for the inverse problem of Electrocardiography in the Sobolev spaces
本研究针对从体表心电图重建心脏电活动的逆问题，利用Sobolev空间中的边值问题求解新进展，证明了模型的唯一性定理，为非侵入性心脏标测的数值方法奠定基础。
On the uniqueness of the quasi-geostrophic equation with the fractional Laplacian
本文研究了带分数阶拉普拉斯算子的表面准地转方程解的唯一性，证明在非齐次Besov空间中无需额外假设即可成立，并提取了方程非线性项良定性所需的最小正则性指标 $s=-1/2$。
We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $Ω\subset\mathbb{R}^n$, let $ρ$ be a quasi-metric on $Ω$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $Ω$. For an open set $Θ\SubsetΩ$, we give sufficient conditions to show that if the local weak Sobolev inequality % \[ \Big(\fint_B |f|^{pσ}dx\Big)^\frac{1}{pσ} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} \] holds for some $σ>1$, all balls $B\subset Θ$, and function…
The complete spectrum of the linearized $p$-Laplacian at a Sobolev extremal
该数学研究确定了尖锐索伯列夫不等式标准径向极值处线性化p-拉普拉斯算子的完整特征基，通过球谐分解等方法实现，索引含角度数与径向模式数。
We prove the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter. This study is motivated by the paper [2] and aims to remove the restriction on dimension imposed there. In this paper, we employ the fixed-point argument developed in [7] to prove the uniquness. Hence, the linearization around the Aubin-Talenti function plays a key role. Furthermore, we need some estimates for the associated perturbed resolvents (see Proposition 3.1).
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of ou…
In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in \cite{FMT1}.
We propose analogues of Green's and Picone's identities for the $p$-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Díaz-Saá inequality for the $p$-sub-Laplacian on stratified Lie groups. Using these we derive a comparison principle and uniqueness of a positive solution to nonlinear hypoelliptic equations on general stratified Lie groups extending to the setting of the general stratified Lie groups previously known results on Euclidean and Heisenberg groups.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities, \begin{align*} (-Δ_p)^su - μ\dfrac{\vert u \vert^{p-2} u}{\vert x \vert^{ps}} = \dfrac{\vert u\vert^{p^*_s(β)-2}u}{\vert x \vert^β} + \dfrac{\vert u\vert^{p^*_s(α)-2} u}{\vert x \vert^α}, \end{align*} where $ x \in \mathbb{R}^N$, $u\in D^{s,p}(\mathbb{R}^N)$, $0<s<1$, $1<p<+\infty$, $N>sp$, $0<α<sp$, $0<β<sp$, $β\neqα$, $μ< μ_H := \inf_{u \in D^{s,p}(\mathbb{R}^N) …
We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields we produce have at least as many integral curves starting from a.e. point of T^d as the number of distinct positive solutions to the continuity equation these vector fields admit. Our work uses convex integration techniques for the continuity equation introduced in [L. Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De Lellis, A…
Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian
该研究针对分数阶拉普拉斯方程基本Dirichlet型问题的非唯一性现象，建立了锐利的唯一性条件，推导了算子的$L^p$估计，引入加权分数阶Sobolev空间并证明其嵌入关系，为相关问题研究提供基础性质支撑。
Uniqueness of positive solutions for m-Laplacian equations with polynomial non-linearity
本文研究含多项式非线性的m-Laplacian方程正解的唯一性，利用第一特征函数的本质唯一性证明p趋于m时的唯一性，并进一步处理Laplacian方程扰动情形的唯一性结果。
In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
We study the sharp constant for the embedding of $W^{1,p}_0(Ω)$ into $L^q(Ω)$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli an…
Inverse problems for the Bakry-Émery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness
本文研究了带边界 Riemannian 流形上 Bakry-Émery Laplacian 两个相关反问题的唯一性与非唯一性，通过 Dirichlet-to-Neumann 映射恢复度量与权重在边界处的 Taylor 系数。
In this paper, we study the minimizing problem: $$ S_{p,1,α,μ}:= \inf_{u\in W^{1,p}(\mathbb{R}^{N})\setminus\{0\}} \frac{ \int_{\mathbb{R}^{N}}|\nabla u|^{p}\mathrm{d}x - μ\int_{\mathbb{R}^{N}} \frac{|u|^{p}}{|x|^{p}} \mathrm{d}x} {\left( \int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}} \frac{|u(x)|^{p^{*}_α}|u(y)|^{p^{*}_α}}{|x-y|^α} \mathrm{d}x \mathrm{d}y \right)^{\frac{p}{2\cdot p^{*}_α}}}, $$ where $N\geqslant3$, $p\in(1,N)$, $μ\in \left[ 0, \left( \frac{N-p}{p} \right)^{p} \right)$, $α\in(0,N)$ and $p^{*}_α= \frac{p}{2}\left(\frac{2N-α}{N-p}\right)$ is the Hardy--Littlewood--Sobolev upper cri…
Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces
本文研究极软势下无角截断的空间均匀玻尔兹曼方程，在带对数修正的临界索伯列夫空间中建立解的存在性、唯一性等，结合费舍尔信息单调性实现解的全局时间延拓，采用相空间与频率二进局域化及精确交换子估计作为核心工具。
In this paper, we investigate the existence of a "weak solutions" for a Neumann problems of $p(x)$-Laplacian-like operators, originated from a capillary phenomena, of the following form \begin{equation*} \displaystyle\left\{\begin{array}{ll} \displaystyle-{\rm{div}}\Big(\vert\nabla u\vert^{p(x)-2}\nabla u+\frac{\vert\nabla u\vert^{2p(x)-2}\nabla u}{\sqrt{1+\vert\nabla u\vert^{2p(x)}}}\Big)=λf(x, u, \nabla u) & \mathrm{i}\mathrm{n}\ Ω,\\ \Big(\vert\nabla u\vert^{p(x)-2}\nabla u+\frac{\vert\nabla u\vert^{2p(x)-2}\nabla u}{\sqrt{1+\vert\nabla u\vert^{2p(x)}}}\Big)\frac{\partial u}{\partialη}=0 & …
Existence and uniqueness of weak solutions to a parabolic nonlocal 1-Laplacian equation
本文研究抛物型非局部1-Laplacian方程，采用Rothe时间离散化方法证明了其弱解的存在性与唯一性，并得出弱解在时间上保持1/2-Hölder连续性的新结论。
In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.
We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued modified log-Sobolev inequality has been obtained. We also show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.
In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations invo…
We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth.
Existence, uniqueness, and decay results for singular $Φ$-Laplacian systems in $\mathbb{R}^N$
本文利用移位法、变分法及多种分析技术，研究了奇异 $\Phi$-Laplacian 系统解的存在性、先验估计、衰减性与唯一性，并给出了 Diaz-Saa 型结果。
Eigenvalue for a problem involving the fractional (p,q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth
本文通过变分法与上下解方法，研究分数阶 (p,q)-Laplacian 奇异椭圆特征值问题，涵盖超临界 Sobolev 增长，并建立解的 L∞ 范数估计。
We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law.
In this paper, we consider the nonlocal elliptic problems in $\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we establish the existence of nonnegative solution. Our results generalize some results obtained by Yang and Wu [Adv. Nonlinear Stud. (2017) \cite{Yang2017}].
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, we derive the existence of positive classical solution to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solution for the fraction Laplacian. Thanks to \cite{XY}, the regularity follows from the boundedness of weak solution.
We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonl…
In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on $\mathbb{S}^n$. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $\mathbb{S}^n$, we first employ the Mobius transform between $\mathbb{S}^n$ and $\mathbb{R}^n$, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $\mathbb{S}^n$ is equivalent to some integral equation in $\mathbb{R}^n$. Then …
Let $Ω\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we build up a global second order regularity estimate $$\|D[|Du|^{p-2} Du]\|_{L^2(Ω)}+\|D[ |\sqrt{A}Du|^{p-2} A Du]\|_{L^2(Ω)} \le C \|f\|_{L^2(Ω)} $$ for inhomogeneous $p$-Laplace type equation \begin{equation} -\mathrm{div}\big(\langle A Du,Du\rangle ^{\frac{p-2}2} A Du\big)=f \quad\rm{in }\ Ω\mbox{ with Dirichlet/Neumann $0$-boundary.} \end{equation} Similar result was also built up for certain bou…
We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.
On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''
本文研究了由局部和非局部算子叠加形成的混合阶算子的临界问题，通过引入具有"错误符号"的分数拉普拉斯算子，实现了非平凡解的产生。
In this work we continue our study initiated in \cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in \cite{GFGP} do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in \cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results
本文研究涉及对数拉普拉斯算子的非局部椭圆方程的存在性、非存在性与唯一性，证明Pohozaev恒等式和Díaz-Saa型不等式，并揭示加权非局部问题的极小能量解在超线性和次线性增长下分别收敛于Brézis-Nirenberg型和逻辑型极限问题的解。
Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential
针对三维及以上有界区域中带 Robin 边界条件的 Schrödinger 算子，本文利用特征值与完整或部分边界 Dirichlet 测量组成的谱数据，研究无界势反演的 Borg-Levinson 唯一性与稳定性。
In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.
Sobolev regularity of the symmetric gradient of solutions to a class of $ϕ$-Laplacian systems
本文研究了ϕ-Laplace系统解的对称梯度的Sobolev正则性，通过构造近似问题并利用均匀高阶可微性估计，证明了在特定Orlicz-Sobolev空间下解的正则性。
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