Extremal solutions for a broad spectrum of nonlinear elliptic systems

Regularity of the extremal solutions associated to elliptic systems

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -Δu =λf(v) \quad \mbox{ in } Ω -Δv =γf(u) \quad \mbox{ in } Ω, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $λ,γ$ are positive parameters, $Ω$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. Assuming $$0<τ_{-}:=\liminf_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq 2…

Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

We consider the semilinear elliptic equation $-Δu =λf(u)$ in a smooth bounded domain $Ω$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $Ω$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabré \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<β_{-}:=\li…

Properties of the extremal solution for a fourth-order elliptic problem

Let $λ^{*}>0$ denote the largest possible value of $λ$ such that $$ \{{array}{lllllll} Δ^{2}u=\fracλ{(1-u)^{p}} & \{in}\ \ B, 0<u\leq 1 & \{in}\ \ B, u=\frac{\partial u}{\partial n} =0 & \{on}\ \ \partial B. {array} . $$ has a solution, where $B$ is the unit ball in $R^{n}$ centered at the origin, $p>1$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough and $1-C_{0}r^{\frac{4}{p+1}}\leq u^{*}(x)\leq 1-r^{\frac{4}{p+1}}$ …

次椭圆算子的第一特征值与基本间隙的极值性质

Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

该研究针对带Dirichlet边界的次椭圆算子，在势函数p-范数约束下探讨第一特征值与基本间隙的极值问题，给出相关理论结果并刻画最优势函数特征。

Regularity of extremal solutions of nonlocal elliptic systems

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& λF(u,v) \qquad \text{in} \ \ Ω, \\ \hfill \mathcal L v &=& γG(u,v) \qquad \text{in} \ \ Ω, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminusΩ, \end{array}\right. \end{eqnarray} with an integro-differential operator, including the fractional Laplacian, of the form \begin{equation*}\label{} \mathcal L(u (x))= \lim_{ε\to 0} \int_{\mathbb R^n\setminus B_ε(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*} when $J$ is a nonnegative…

Regularity of minimizers of semilinear elliptic problems up to dimension four

We consider the class of semi-stable solutions to semilinear equations $-Δu=f(u)$ in a bounded smooth domain $Ω$ of $R^n$ (with $Ω$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n \leq 4$, we establish an priori $L^\infty$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $Ω$ is convex. This result was previously known only in dimensions $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n \leq 9$ the boun…

Extremal solution and Liouville theorem for anisotropic elliptic equations

We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left\{ \begin{aligned} -Qu&=λe^{u} \quad \mbox{in}\quadΩ\\ u&=0 \quad \mbox{on}\quad\partialΩ,\\ \end{aligned} \right. \end{equation} where $λ>0$ is a parameter, $Ω\subset\mathbb{R}^{N}$ with $N\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)). $$ Here, $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $…

On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation $βΔ^2 u-τΔu=\fracλ{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=Δu=0$ on $\partial B$, where $λ>0$ is a parameter, while $τ>0$, $β>0$ are fixed constants. It is known that there exists a $λ^{*}$ such that for $λ>λ^{*}$ there is no solution while for $λ<λ^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and $β, τ>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$, $β>0$, and $τ…

Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

We consider the fourth order problem $Δ^{2}u=λf(u)$ on a general bounded domain $Ω$ in $R^{n}$ with the Navier boundary condition $u=Δu=0$ on $\partial Ω$. Here, $λ$ is a positive parameter and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. Let $$0<τ_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $λ_{m}$ sati…

Lyapunov type inequality for extremal Pucci's equations

In this article, we establish Lyapunov type inequality for the following extremal Pucci's equation \begin{equation*} \left\{ \begin{aligned}{} \mathcal{M}^{+}_{λ,Λ}(D^{2}u)+a(x)u&=0~\text{in}~Ω,\\ u&=0~\text{on}~\partialΩ, \end{aligned} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\R^{N},~N\geq2$. This works generalize the well-known works on Lyapunov inequalities to fully nonlinear elliptic equations.

全次极端范围 $|a|<M$ 内 Kerr 线性化引力扰动的椭圆曲率估计

Elliptic curvature estimates for linearised gravitational perturbations of Kerr in the full sub-extremal range $|a|<M$

本文针对 Kerr 外部解周围的线性化真空 Einstein 方程，提出证明线性化曲率量椭圆 $L^2$ 估计的方案，适用于全次极端参数范围 $|a|<M$。

含Pucci极值算子的PDE问题粘性解的表示公式

Representation Formula for Viscosity Solution to a PDE Problem involving Pucci's Extremal Operator

本研究借助源自Denis、Hu与Peng（2010）的动态规划原理，推导了含Pucci极值算子的椭圆Dirichlet问题粘性解的表示公式，该公式是Feynman-Kac公式的非线性推广。

A new approach to Liouville theorems for elliptic inequalities

In this article we review a new method for proving the nonexistence of positive solutions of elliptic inequalities in unbounded domains in $\rn$, which was recently introduced by the authors. We expose our method and new results on the two most frequently encountered cases: inequalities involving the p-Laplacian or the Pucci extremal operator.

关于一类带对数项的超临界 Hardy-Sobolev 型不等式及相关极值问题

On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

本文研究带对数项的超临界 Hardy-Sobolev 型不等式，证明紧性丧失情形下极值函数存在，并用于确立一类相关椭圆偏微分方程弱解的存在性。

Regularity of stable solutions to semilinear elliptic equations on Riemannian models

We consider the reaction-diffusion problem $-Δ_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian manifolds includes the classical \textit{space forms}, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish $L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend on the nonlinearity $f$. As an application, under standard assumptions on …

Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& λg(x) f(u) \qquad \text{in}\ Ω\hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ λ>0$ is a parameter and $ Ω$ is a smooth bounded domain in $ \IR^N$, $ N \ge 2$. Here $ g$ is a positive function and $ f$ is an increasing, convex function with $ f(0)=1$ and either $ f$ blows up at 1 or $ f$ is superlinear at infinity. We show that the extremal solution $u^*$ associated with the extremal parameter $ λ^*$ is the unique solution. We also show that when $f$ is suitably supercritical and $ Ω$ s…

Behavior near the origin of $f'(u^\ast)$ in radial singular extremal solutions

Consider the semilinear elliptic equation $-Δu=λf(u)$ in the unit ball $B_1\subset \mathbb{R}^N$, with Dirichlet data $u|_{\partial B_1}=0$, where $λ\geq 0$ is a real parameter and $f$ is a $C^1$ positive, nondecreasing and convex function in $[0,\infty)$ such that $f(s)/s\rightarrow\infty$ as $s\rightarrow\infty$. In this paper we study the behavior of $f'(u^\ast)$ near the origin when $u^\ast$, the extremal solution of the previous problem associated to $λ=λ^\ast$, is singular. This answers to an open problems posed by Brezis and Vázquez.

Large critical exponents for some second order uniformly elliptic operators

In this paper we investigate the critical exponents of two families of Pucci's extremal operators. The notion of critical exponent that we have chosen for these fully nonlinear operators whihc are not variational is that of threshold betweeen existence and nonexistence of the solutions for semilinear equations with pure power nonlinearities. Interesting new exponents appear in this context.

Boundary regularity for viscosity solutions of fully nonlinear elliptic equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is $C^{1,α}$ on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is $C^{2,α}$ on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.

On a class of fully nonlinear elliptic equation in dimension two

We study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover for the $\mathcal{M}^-_{λ,Λ}$ operator, we show the existence of a critical exponent and give bounds for it.

奇异椭圆问题的极值参数、多重性与不存在性

Extremal parameters, multiplicity and nonexistence for a singular elliptic problem

本文研究一类奇异椭圆问题，确定了存在非平凡非负弱解的尖锐阈值，并证明在阈值左侧邻域内至少存在两个不同解，且其能量符号与先前结果不同。

The critical dimension for a fourth order elliptic problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0<u_\la<1$ exists for $\la\in (0,\la^*)$, while there is none of any kind when $\la>\la^*$. Our main result asserts that the extremal solution $u_{…

Extremals For Fractional Order Hardy-Sobolev-Maz'ya Inequality

In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.

关于 Caffarelli-Kohn-Nirenberg 不等式极值元的分类

On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

本文研究 Caffarelli-Kohn-Nirenberg 不等式对应的临界椭圆方程，在无有限能量假设且内蕴维度 $n\in(\frac{3}{2},5]$ 时，分类了 $\mathbb{R}^d$ 凸锥中的正解。

Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of "A selection principle for the sharp quantita…

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bel…

Liouville-type theorems for fully nonlinear elliptic equation in half spaces

In \cite{LWZ}, we establish Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in \cite{LWZ} to investigate the role of boundedness assumption in proving Liouville-type theorems for fully nonlinear equations. We remove the boundedness assumption of solutions which was required in the proof of Liouville-type theorems for fully nonlinear elliptic equations or systems in half spaces. We also prove the Liouville-type theorems for supersolutions of…

The solvability of Brezis-Nirenberg type problems of singular quasilinear elliptic equation

In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…

具有可变退化性和Hamilton低阶项的全非线性方程的梯度正则性

Gradient Regularity for Fully Nonlinear Equations with Variable Degeneracy and Hamiltonian Lower-Order Terms

研究具有可变梯度退化和Hamilton型低阶项的全非线性椭圆方程的粘性解，通过紧致性估计和扁平化改进迭代，证明了内梯度Hölder估计、源项和Hamilton系数消失点的逐点改进以及极值点的Schauder型估计。

Bounds for the extremal parameter of nonlinear eigenvalue problems and application to the explosion problem in a flow

We consider the nonlinear eigenvalue problem $ L u = λf(u) $, posed in a smooth bounded domain $ Ω\subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential operator, $ λ> 0 $ and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing and convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. First we present some upper and lower bounds for the extremal parameter $ λ^{*} $ and the extremal solution $ u^{*} $. Then we apply the results to the operator $ L_A = - Δ+ A…

Some remarks on biharmonic elliptic problems with a singular nonlinearity

We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} Δ^{2}u=\fracλ{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right. %\eqno(M_λ) $$ where $\B$ is the unit ball in $\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $λ^{*}>0$ such that for $λ\in (0,λ^{*})$ there exists a minimal (classical) solution $\underline{u}_λ$, which satisfies $0<\underline{u}_λ<1$. In the extremal case $λ=λ^{*}$, we prove the existence of a weak solution which is unique solution even in a v…

Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the $p$-Laplacian and application

We derive a priori bounds for positive supersolutions of $ - Δ_{p} u = ρ(x) f(u) $, where $p>1$ and $Δ_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem $ - Δ_{p} u = λf(u) $, with Dirichlet boundary condition, where $ f $ is a nondecreasing continuous differentiable function on $[0,\infty]$ such that $ f(0) > 0 $, $ f(t)^{\frac{1}{p-1}} $ is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter $ λ_{p}^{*} $. In particular, we conside…

The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material

We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $λ$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $λ_0^*$ for which the local minimum has non-negative energy when $λ\ge λ_0^*$. Moreover we characterize the extremal parameter $λ^*$ for which if $λ>λ^*$, then the only solution to the Kirchhoff equat…

带α齐次权的Sobolev临界嵌入极值函数的存在性

Existence of an extremal function of Sobolev critical embedding with an $α$-homogeneous weight

本研究针对此前提出的带单项式权的临界Sobolev型嵌入极值函数存在性问题给出肯定答案，并将结果推广至更广泛的α齐次权情形。

混合全非线性局部和非局部椭圆算子在海森堡群中

Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group

本文研究了在海森堡群中混合全非线性局部和非局部椭圆算子的粘性解的存在性、正则性和比较原理。

弱耦合系统Pucci极值算子的解的存在性和多重性结果

Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator

本文研究了由Pucci极值算子支配的弱耦合非线性椭圆方程组的多个正解的存在性，通过上下解方法、固定点论证和分支技术，证明了参数μ足够大时解的存在性和多重性。

$W^{2,p}$ interior estimates of fully nonlinear elliptic equations

In this paper, we generalize the $W^{2,p}$ interior estimates of fully nonlinear elliptic equations that were obtained by Caffarelli in [1]. The generalizations are carried out in two directions. One is that we relax the regularity requirement on the "constant coefficients" equations and the other one is that we broaden the range of $p$.