Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embeddi…

Gradient estimates via two-point functions for elliptic equations on manifolds

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifol…

Fractional Elliptic problem with Finite many critical Hardy--Sobolev Exponents

In this paper, we consider the following problem: $$ (-Δ)^{s} u -\frac{ζu}{|x|^{2s}} = \sum_{i=1}^{k} \frac{|u|^{2^{*}_{s,θ_{i}}-2}u} {|x|^{θ_{i}}} , \mathrm{~in~} \mathbb{R}^{N}, $$ where $N\geqslant3$, $s\in(0,1)$, $ζ\in \left[ 0,4^{s}\frac{Γ(\frac{N+2s}{4})}{Γ(\frac{N-2s}{4})} \right)$, $2^{*}_{s,θ_{i}}=\frac{2(N-θ_{i})}{N-2s}$ are the critical Hardy--Sobolev exponents, the parameters $θ_{i}$ satisfy a suitable assumption. By using Morrey space, refinement of Hardy--Sobolev inequality and variational method, we establish the existence of nonnegative solution. Our result generalizes the resu…

具有奇异初值的半线性抛物方程组解的存在性

Existence of solutions for a semilinear parabolic system with singular initial data

本文研究具有奇异初值的半线性抛物方程组，利用一致局部 Morrey 空间和一致局部弱 Zygmund 型空间，建立了保证解存在的关于初值的精确充分条件。

带单项权的Sobolev不等式与等周不等式

Sobolev and isoperimetric inequalities with monomial weights

我们考虑$\mathbb R^n$中的单项权$|x_1|^{A_1}...|x_n|^{A_n}$，其中$A_i\geq0$对每个$i=1,...,n$都是实数，并建立了包含这个权的Sobolev不等式、等周不等式、Morrey不等式和Trudinger不等式.它们是经典的类似物，其中Lebesgue测度$dx$被$|x_1|^{A_1}...|x_n|^{A_n}dx$取代，并且它们包含最佳或临界指数（取决于$A_1$，. $A_n$）。更重要的是，对于Sobolev不等式和等周不等式，我们得到了最佳常数和极值函数。当$A_i$是非负的\textit{integers}时，这些不等式在$\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$中的轴对称函数和域上写时，正好是欧几里得空间$\mathbb R^D$（没有权）中的经典不等式。

Weighted anisotropic Sobolev inequality with extremal and associated singular problems

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0…

临界各向异性 $p$-Laplace 方程：无有限能量情形的分类与 Schoen 型 Harnack 不等式

Critical anisotropic $p$-Laplace equations without finite energy: classification and a Schoen-type Harnack inequality

本文在无有限能量假设下完整分类了临界各向异性 p-Laplace 方程的非负局部弱解，证明所有非零解均为各向异性 Sobolev 气泡，并建立了 Schoen 型 Harnack 不等式。