Remarks on Eigenvalue Problems for Fractional $p(\cdot)$-Laplacian

Global existence and boundedness in a chemotaxis-Stokes system with slow $p$-Laplacian diffusion

This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion \begin{equation}\nonumber \left\{ \begin{aligned} &n_t+u\cdot\nabla n=\nabla\cdot\left(|\nabla n|^{p-2}\nabla n\right)-\nabla\cdot(n\nabla c), &x\inΩ,\ t>0,\ \ &c_t+u\cdot\nabla c=Δc-nc,&x\inΩ,\ t>0,\ \ &u_t=Δu+\nabla P+n\nablaϕ,&x\inΩ,\ t>0,\ \ &\nabla\cdot u=0, &x\inΩ,\ t>0,\ \ \end{aligned} \right. \end{equation} where $ϕ\in W^{2,\infty}(Ω)$ is the gravitational potential. It is proved that global bounded weak solutions …

A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity

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 u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0…

半正$p(\cdot)$-拉普拉斯问题正解的存在性

Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem

本文针对带次临界增长非线性项的半正$p(\cdot)$-拉普拉斯边值问题，在Ambrosetti-Rabinowitz型条件下，结合山路引理等工具证明了参数足够小时正弱解的存在性。

一类含权的退化 Kirchhoff 型变 $s(\cdot)$ 阶分数阶 $p(\cdot)$-Laplacian 问题

A degenerate Kirchhoff-type problem involving variable $s(\cdot)$-order fractional $p(\cdot)$-Laplacian with weights

本文针对含权的非局部变阶分数阶 $p(\cdot)$-Kirchhoff 方程，在适当结构条件下证明非平凡解的存在性与多重性，并将结论推广至退化情形。

Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion

This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(nχ(c)\nabla c),& x\inΩ,\ t>0, c_t+u\cdot\nabla c=Δc-nf(c),& x\inΩ,\ t>0, u_t+(u\cdot\nabla) u=Δu+\nabla P+n\nablaΦ,& x\inΩ,\ t>0, \nabla\cdot u=0,& x\inΩ,\ t>0 \end{array}\right. \end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Here, $Φ\in …

Some Remarks on Robin-Laplacian Eigenvalues

We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.

A New Proof of the Equivalence of Weak and Viscosity Solutions to the Homogeneous $\texttt{p}(\cdot)$-Laplacian in $\mathbb{R}^n$

We present a new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$. The proof of the equivalence in the variable exponent case in Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010) and extended the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}$-Laplace equation in $\mathbb{R}^n$, given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponent case and the variable exponent case, the main argument is based on th…

A Free boundary problem for the $p(x)$- Laplacian

We consider the optimization problem of minimizing $\int_Ω|\nabla u|^{p(x)}+ λχ_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(Ω)$ with $u-ϕ_0\in W_0^{1,p(\cdot)}(Ω)$, for a given $ϕ_0\geq 0$ and bounded. $W^{1,p(\cdot)}(Ω)$ is the class of weakly differentiable functions with $\int_Ω|\nabla u|^{p(x)} dx<\infty$. We prove that every solution $u$ is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, $Ω\cap\partial\{u>0\}$, is a regular surface.

Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications

We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.

A remark on nonlocal Neumann conditions for the fractional Laplacian

We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.

关于具有夹层型和任意增长的Kirchhoff型p(·)-Laplacian问题

On Kirchhoff-type p(.)-Laplacian problems with sandwich-type and arbitrary growth

针对具有任意增长和夹层型增长的Kirchhoff型p(·)-Laplacian问题，通过截断论证和先验估计，证明了正有界弱解的存在性。

The Spatially Variant Fractional Laplacian

We introduce a definition of the fractional Laplacian $(-Δ)^{s(\cdot)}$ with spatially variable order $s:Ω\to [0,1]$ and study the solvability of the associated Poisson problem on a bounded domain $Ω$. The initial motivation arises from the extension results of Caffarelli and Silvestre, and Stinga and Torrea; however the analytical tools and approaches developed here are new. For instance, in some cases we allow the variable order $s(\cdot)$ to attain the values $0$ and $1$ leading to a framework on weighted Sobolev spaces with non-Muckenhoupt weights. Initially, and under minimal assumptions,…

The obstacle problem for the fractional Laplacian with critical drift

We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{(-Δ)^s u + b \cdot \nabla u,\,u -φ\right\} = 0$ in $\mathbb{R}^n$, in the critical regime $s = \frac{1}{2}$. Our main result establishes the $C^{1,α}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \[ u(x)-φ(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tildeγ(x_0)} + o\left(|x-x_0|^{1+\tildeγ(x_0)+σ}\right), \] \[ \tildeγ(x_0) = \frac{1}{2}+\frac{1}π \arctan (b\cdot e), \] where $e \in \mathbb{S}^{n-1}$ is the normal vector to the free boundary, $σ>0$, and $c_0> 0$. We al…

关于 p-Laplacian 比较原理并推广至 (p,q)-Laplacian 的评注

Remarks on comparison principles for p-Laplacian with extension to (p,q)-Laplacian

本文将 p-Laplacian 算子的比较原理推广到含 (p,q)-Laplacian 的拟线性方程，证明加入 q-Laplacian 可弱化低阶 Hamiltonian 的假设，并给出多项式增长情形。

Remarks on regularity for $p$-Laplacian type equations in non-divergence form

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^γ\left(Δu+(p-2)Δ_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ Ω.$$ We investigate local $C^{1,α}$ regularity of viscosity solutions in the full range $γ>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $γ$ is close to 0.

Remarks on Local Boundedness and Local Hölder Continuity of Local Weak Solutions to Anisotropic $p$-Laplacian Type Equations

Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.

Remarks on blow-up phenomena in p-Laplacian heat equation with inhomogeneous nonlinearity

We investigate the $p-$Laplace heat equation $u_t-Δ_p u=ζ(t)f(u)$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$. Using differential inequalities arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial data $u_0$. We also give an upper bound for the blow-up time in each case.

p(·)-Laplace算子的有限元近似

Finite element approximation of the $p(\cdot)$-Laplacian

本文研究了p(·)-Laplace方程的Dirichlet问题的先验估计，证明了当p为α-Hölder连续时，有限元近似解的梯度以O(h^α)速率收敛。

A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω\subset\mathbb{R}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} λ_1(Ω)-λ_1(B)\geq c\left(P(Ω)-P(B) \right)^{2}, \end{equation*} where by $λ_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present pap…

Effect of Non-linear Lower Order Terms in Quasilinear Equations Involving the $p(x)$-Laplacian

In this work, we study the existence of $W_0^{1, p(\cdot)}$-solutions to the following boundary value problem involving the $p(\cdot)$-Laplacian operator: \begin{equation*} \left\lbrace \begin{array}{l} -Δ_{p(x)}u+|\nabla u|^{q(x)}=λg(x)u^{η(x)}+f(x), \quad\textnormal{ in } Ω, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq 0, \quad\textnormal{ in } Ω \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad \text{on } \partialΩ. \end{array} \right. \end{equation*}under appropriate ranges on the variable exponents. We give assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $η$ u…

无界区域中的嵌入结果与分数阶 $p(x,.)$-Laplacian 问题

Embedding Results and Fractional $p(x,.)$-Laplacian Problem in Unbounded Domains

本文建立无界区域上变指数分数阶 Sobolev 空间的新连续嵌入定理，并据此用变分方法证明分数阶 $p(x, \cdot)$-Laplacian 非局部椭圆问题存在非平凡弱解。

Remarks on the validity on the maximum principle for the $\infty$-Laplacian

In this note we give three counter-examples which show that the Maximum Principle generally fails for classical solutions of a system and a single equation related to the $\infty$-Laplacian. The first is the tangential part of the $\infty$-Laplace system and the second is the scalar $\infty$-Laplace equation perturbed by a linear gradient term. The interpretations of the Maximum Principle for the system are that of the Convex Hull Property and also of the Maximum Principle of the modulus of the solution.

Some Remarks on the Fucik Spectrum of the p-Laplacian and Critical Groups

We compute critical groups of variational functionals arising from quasilinear elliptic boundary value problems with jumping nonlinearities, when the asymptotic limits of the equation lie in various regions of the plane that are separated by certain curves of the Fucik spectrum. As an application some existence and multiplicity results are established via Morse theoretic and perturbation arguments.

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=Δ^{α/2}+ b\cdot\nabla$ with $α\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.

Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $α$ in the boundary condition, or equivalently of the volume $M$. We obtain …

Some remarks on the homogeneous Boltzmann equation with the fractional Laplacian term

We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^α$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we o…

Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion

Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regul…

关于论文的评论：Ikehata, M., 方程 $\nabla\cdot(σ-iωε)\nabla u=0$ 的逆边界值问题的提取公式, Inverse Problems, 18(2002), 1281-1290

Remarks on the paper: Ikehata, M., Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$, Inverse Problems, 18(2002), 1281-1290

本文针对Ikehata关于逆边界值问题提取公式的论文，给出了其中定理1.1与1.2跳跃条件的评论及其推论。

Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

This paper considers the inverse boundary value problem for the equation $\nabla\cdot(σ\nabla u+a|\nabla u|^{p-2}\nabla u)=0$. We give a procedure for the recovery of the coefficients $σ$ and $a$ from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

Remarks on an optimization problem for the $p-$Laplacian

In this note we give some remarks and improvements on a recent paper of us [3] about an optimization problem for the $p-$Laplace operator that were motivated by some discussion the authors had with Prof. Cianchi.

具有对数非线性项的$p$-Laplacian方程正规化解的存在性与集中性

Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity

本文用变分法研究带对数非线性项的p-Laplacian方程正规化解的存在性与集中性，证明解的数量取决于势函数轮廓，半经典极限下解集中于全局最小点，并推广到质量超临界情形。

Some remarks on the Krein--von Neumann extension of different Laplacians

We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $n\in \mathbb N$ infinitely many elements of the class have $n$-dimensional null space.

关于一类特殊Finsler $p$-Laplacian方程的注记

Remark on a special class of Finsler $p$-Laplacian equation

本文研究一类特殊Finsler $p$-Laplacian方程，指出在$(H_M)$条件下Finsler $p$-Laplacian与经典$p$-Laplacian经线性变换等价，为已有文献的核心结论提供了更直接的推导路径，填补了相关文献的表述空白。

Solvability of Doubly Nonlinear Parabolic Equation with $p$-Laplacian

In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However…

Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory

The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem \[ \left\{\begin{array}{lr} - \divg (A(x,u)\, |\nabla u|^{p-2}\, \nabla u) + \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\ f(x,u) & \hbox{in $Ω$,}\\ u\ = \ 0 & \hbox{on $\partialΩ$,} \end{array} \right. \] where $Ω\subset \R^N$ is a bounded domain, $N\ge 2$, $p > 1$, $A$ is a given function which admits partial derivative $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $f$ is asymptotically $p$-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if $A(…

A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - ν(-\triangle)^α u b_t + u \cdot \nabla b & = & b \cdot \nabla u - κ(-\triangle)^β b \end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslantα< 1 / 2,\,\, β\geqslant 1, \,\,3α+ 2β>3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $α=0$ and $β>\frac 32$.

Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation

We study the two-dimensional generalized magnetohydrodynamics system with generalized dissipation and diffusion in terms of fractional Laplacians. It is known that the classical magnetohydrodynamics system with full Laplacians in both dissipation and diffusion terms admits a unique global strong solution pair. Making use of the special structure of the system in the two-dimensional case, we show in particular that the solution pair remains smooth when we have zero dissipation but only magnetic diffusion with its power of the fractional Laplacian $β> \frac{3}{2}$.

System of Degenerate Parabolic $p$-Laplacian

In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla \bold{u}\right), \qquad \qquad \left(p>2\right). \end{equation*} More precisely, we show the uniqueness and existence of solution $\bold{u}$ and investigate a priori $L^{\infty}$ boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component $u^l$, $\left(1\leq l\leq k\right)$, converges to the functi…

On a quasilinear elliptic problem involving the 1-laplacian operator and a discontinuous nonlinearity

In this work, we study a quasilinear elliptic problem involving the 1-laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - β)$. Our approach is based on an analysis of the associated p-laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $β\to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

A Harnack inequality for weak solutions of the Finsler $γ$-Laplacian

We study regularity of the Finsler $γ$-Laplacian, a general class of degenerate elliptic PDEs which naturally appear in anisotropic geometric problems. Precisely, given any strictly convex family of $C^{1}$-norms $\{ ρ_{x}\}$ on $\mathbb{R}^{n}$ and $γ> 1$, we consider the $W^{1,γ}(Ω)$ solutions of the anisotropic PDE $$ \displaystyle \int_Ω \left \langle ρ_{x}(Du)^{γ-1} (D ρ_{x})(Du), D φ\right \rangle = \int_Ω \vec{F} \cdot D φ+ f φ\qquad \forall φ\in W^{1,γ^{\prime}}_{0}(Ω). $$ Under the mild assumption $|ξ|^{-1} ρ_{x}( ξ) \in [ν, Λ]$ for all $(x,ξ) \in Ω\times \mathbb{R}^{n}$ and some $0 <…

Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.

一类带有两个参数和变阶分数阶 $p(\cdot)$-Laplacian 的新型多重非局部问题

A new class of multiple nonlocal problems with two parameters and variable-order fractional $p(\cdot)$-Laplacian

本文研究了一类涉及变阶p(x)-分数阶Laplacian的新型三重非局部Kirchhoff问题，结合变分法及Fountain定理与对称山路定理，分别在两种参数条件下证明了该问题解的存在性与多重性。

Holder continuity of weak solutions of p-Laplacian PDE's with VMO coefficients

We consider solutions $u\in W^{1,p}\big(Ω;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\inΩ\subseteq\mathbb{R}^{n}$, where $Ω$ is open and bounded. More generally, we consider solutions of the elliptic system \begin{equation} \nabla\cdot\left(a(x)g'\big(a(x)|Du|\big)\frac{Du}{|Du|}\right)=0\text{, }x\inΩ\notag \end{equation} as well as minimizers of the functional \begin{equation} \int_Ωg\big(a(x)|Du|\big)\ dx.\notag \end{equation} In each case, the coefficient map $a\ : \ Ω\rightarrow\mathbb{R}$ is only assu…

Liouville theorems for infinity Laplacian with gradient and KPP type equation

In this paper, we prove new Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ where $γ\in [0, 2]$ and $Δ^γ_\infty$ is a $(3-γ)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-γ}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-γ}$ decreasing in $s$, we further obtain the u…

On positive solutions of p-Laplacian-type equations

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_Ω. (|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.

Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

This paper deals with the eigenvalue problem for the operator $L=-Δ-x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $λ_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N} $ the following minimization problem $$ \min\left\{λ_k(Ω): \> Ω\>\mbox{quasi-open} \>\mbox{set}, \> \int_Ωe^{|x|^2/2}dx\le c\right\} $$ has a solution.

The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.

Robin边界条件下带漂移项拉普拉斯算子的Faber-Krahn不等式

A Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition

本研究证明了大β参数下Robin边界条件带漂移拉普拉斯算子的Faber-Krahn不等式，并论证了满足L^∞约束、使对应算子主本征值取极值的漂移的存在唯一性。

Remarks on functions with bounded Laplacian

$Δψ:=\frac{\partial^2 ψ}{\partial x_1^2}+\frac{\partial^2 ψ}{\partial x_2^2}$ being locally bounded does not imply that $D^2ψ$ is locally bounded. However, we prove that if $ψ$ is invariant under rotation by $\frac{2π}{m}$, for some $m\geq 3$, and $Δψ$ is locally bounded, then $$\sup_{x\in B_1(0)}\frac{|\nabla ψ(x)|}{|x|}<\infty.$$ This is sharp in that there are examples of functions $ψ$ for which $Δψ$ is locally bounded, which are invariant under rotation by $π$ with $|ψ(x)-ψ(0)|\approx |x|^2 |\log|x||$ as $|x|\rightarrow 0$. This bound and its generalizations could be of use in different co…

Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities

In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} \;\;\; \begin{array}{rl} (-Δ)_{p(\cdot)}^{s} u&=λ~ a(x)| u|^{q(x)-2}u+\frac{α(x)}{α(x)+β(x)}c(x)| u|^{α(x)-2}u| v| ^{β(x)},\hspace{2mm} x\in Ω; \\ (-Δ)_{p(\cdot)}^{s} v&=μ~ b(x)| v|^{q(x)-2}v+\frac{α(x)}{α(x)+β(x)}c(x)| v|^{α(x)-2}v| u| ^{β(x)},\hspace{2.5mm} x\in Ω; \\ u=v&=0 ,\hspace{1cm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $Ω\subset\mathbb R^N,~N\geq2$ is …

Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying Hörmander's condition and ${Δ_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ${Δ_L}$ and $Δ_L^2$. These inequalities extend Yang's inequalities for Dirichlet eigenvalues of Laplacian to the settings here and the forms of inequalities are more general than Yang's inequalities. To obtain them, we give a generalization of the inequality by Chebyshev.

Multiplicity and uniform estimate for a class of variable order fractional $p(x)$-Laplacian problems with concave-convex nonlinearities

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+\left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is continuous function with…

Nonlinear pseudo-differential equations defined by elliptic symbols on ${\lp}$ and the fractional Laplacian

We develop an $L^p(\mathbb{R}^n)$-functional calculus appropriated for interpreting "non-classical symbols" of the form $a(-Δ)$, and for proving existence in $L^q(\mathbb{R}^n)$, some $q > p$, of solutions to nonlinear pseudo-differential equations of the form $[1 + a(-Δ)]^{s/2} (u) = V(\cdot, u)$. More precisely, we use the theory of Fourier multipliers for constructing suitable domains on which the formal operator appearing in the above equation can be rigorously defined, and we prove existence of solutions belonging to these domains. We also include applications of the theory to equations o…

拉普拉斯算子非正则扰动的定量唯一延拓性

Quantitative unique continuation for non-regular perturbations of the Laplacian

本文针对$d≥3$维空间中拉普拉斯算子非正则扰动的椭圆方程，通过改进卡尔曼估计并结合Wolff的论证方法，建立了基于势函数范数的唯一延拓性定量估计。