We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u) = w(|x|)g(u)$ on the Euclidean space $R^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behavior at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behavior at infinity of the growth rate function of $w$.
In this article we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Our results complete and improve a recently work published by Zhang and Zhou (Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights, Applied Mathematics Letters, Volume 50, December 2015, Pages 48--55).
We characterize semistable radial solutions of the equation $S_k\left(D^2u\right)=g(u)\;\mbox{in } B_1$, where $B_1$ is the unit ball of $\mathbb{R}^n$, $D^2u$ is the Hessian matrix of $u,\,g$ is a positive $C^1$ nonlinearity and $S_k\left(D^2u\right)$ denotes the $k$-Hessian operator of $u$. This class of radial solutions has been recently introduced by the authors in [8]. The proofs are new relative to those given in [8] and focus on the structure of the equation directly, thereby improving some previous results.
In this paper we deduce a formula for the fractional Laplace operator $(-Δ)^{s}$ on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with $(-Δ)^{s}$, and apply it to a problem related to the Hessian inequality of Sobolev type: $$\int_{\mathbb{R}^n}|(-Δ)^{\frac{k}{k+1}} u|^{k+1} dx \le C \int_{\mathbb{R}^n} - u \, F_k[u] \, dx, $$ where $F_k$ is the $k$-Hessian operator on $\mathbb{R}^n$, $1\le k < \frac{n}{2}$, under some restrictions on a $k$-convex function $u$. In particular, we show that the class of $u$ for which the above inequal…
In this paper, the existence and multiplicity of nontrivial radial convex solutions to general coupled system of $k_i$-Hessian equations in a unit ball are studied via a fixed-point theorem. In particular, we obtain the uniqueness of nontrivial radial convex solution and nonexistence of nontrivial radial $\boldsymbol{k}$-admissible solution to a power-type system coupled by $k_i$-Hessian equations in a unit ball. Moreover, using a generalized Krein-Rutman theorem, the existence of $\boldsymbol{k}$-admissible solutions to an eigenvalue problem in a general strictly $(k-1)$-convex domain is also…
We consider the problem \begin{equation}(1)\;\;\; \begin{cases} S_k(D^2u)= λ|x|^σ (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $λ>0$, $q > k$ and $σ\geq 0$. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generali…
In this paper we prove Holder regularity of the derivative of radial solutions to fully nonlinear equations when the operator is hessian, homogenous of degree 1 in the Hessian, homogenous of some degree $α>-1$ in the gradient and which is elliptic when the gradient is not null.
We are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }Ω,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\in C^2(Ω)$ ($1\leq k\leq N$) and $Ω\subset\mathbb{R}^N$ $(N\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\geq 0$, $p\geq s\geq 0$ and $(k-m)(k-s)\neq pq$. In the case where $Ω$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. …
In this paper, we study negative classical solutions and stable solutions of the following $k$-Hessian equation $$ F_k(D^2V)=(-V)^p \quad in~R^n $$ with radial structure, where $n \geq 3$, $1<k<n/2$ and $p>1$. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research $k$-Hessian equations without radial structure.
We derive a priori interior Hessian estimates for semiconvex solutions to the sigma-2 equation. An elusive Jacobi inequality, a transformation rule under the Legendre-Lewy transform, and a mean value inequality for the still nonuniformly elliptic equation without area structure are the key to our arguments. Previously, this result was known for almost convex solutions.
In this paper, we study the Dirichlet problem of Hessian quotient equations in exterior domains. By estimating the eigenvalues of the solution, the necessary and sufficient conditions on existence of radial solutions are obtained. Applying the solutions of ODE, the viscosity subsolutions and supersolutions are constructed and then the existence of viscosity solutions for exterior problems is established by the Perron's method.
Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University（北京大学数学科学学院纯数学与应用数学重点实验室）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
In this paper, we study the Hessian equation with infinite Dirichlet (blow-up) boundary value conditions. Using radial functions and techniques of ordinary differential inequality, we construct various barrier functions (super-solution and sub-solution). Existence and non-existence theorems are proved by those barriers, maximum principle and theory of viscous solutions. Furthermore, generic boundary blow-up rates for the solutions are derived.
In this note, we prove some non-existence results for Dirichlet problems of complex Hessian equations. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don't know if a priori radially symmetric property holds in the complex case.
This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix…
In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze …
In this paper, we establish a curvature estimate for semi-convex solutions of Hessian equations in hyperbolic space. We also obtain a curvature estimate for admissible solutions to prescribed curvature measure type problem in hyperbolic space. A crucial ingredient in both estimates is a concavity inequality for Hessian operator.
Interior Hessian estimates for Hessian quotient equations in dimension three
School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院与高等研究院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular …
We study the long-time behavior of solutions of the $k$-Hessian evolution equation $u_t=S_{k}(D^2 u)$, posed on a bounded domain of the $n$-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of $u$ is precisely described by this special solution. Further, we initiate the study of that dynamic phenomenon on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call $k$-Barenblatt solutions. These solutions present some common properties as those of well…
We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equati…
In this paper, we consider a kind of $k$-Hessian type equations $S_k^{\frac{1}{k}}(D^2u+μ|D u|I)= f(u)$ in $\mathbb{R}^n$, and provide a necessary and sufficient condition of $f$ on the existence and nonexistence of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $μ$ respectively, which embrace the standard Hessian equation case ($μ=0$) by Ji and Bao (Proc Amer Math Soc 138: 175--188, 2010) as a typical example. The difference between the semilinear case ($k=…
Interior Hessian Estimates for Semi-convex Solutions of the $σ_2/σ_1$ Equation with Lipschitz Right-Hand Sides
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for degenerate, (and non-degenerate), augmented Hessian equations, with strictly regular associated matrix functions. By establishing local second derivative estimates at the boundary and proving viscosity comparison principles, we show that the solution is correspondingly smooth near boundary points where the appropriate uniform convexity is satisfied.
The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= λρ(|x|) (1-u)^q &\mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B. \end{cases} \end{equation} Here, $B$ denotes the unit ball in $\mathbb R^n\!$, $n>2k$, $λ$ is a positive parameter and $q>k$ with $k\in \mathbb N$. The function $rρ'(r)/ρ(r)$ satisfies very general conditions in the radial direction $r=|x|$. We show the existence, nonexistence, and multiplicity of solutions to Problem \eqref{Eq:Ma…
We study radial symmetry of large solutions of the semi-linear elliptic problem Δu + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.
We consider the problem \begin{equation}\label{Eq:Abstract} (1)\;\;\;\begin{cases} S_k(D^2u)= λ(1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $λ>0$ and $q > k$. We study the existence of negative bounded radially symmetric solutions of (1). In the critical case, that is when $q$ equals Tso's critical exponent $q=\frac{(n+2)k}{n-2k}=:q^*(k)$, we obtain exactly either one or two solutions depending on the parameters. Further, we express such solutions expl…
This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-Δu=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We establish sharp pointwise estimates for such solutions. In addition, we prove that in dimension $N=2$, any semi-stable radial weak solution of $-Δu=f(u)$, posed in $B_1$ with Dirichlet data $u|_{\partial B_1}=0$, is regular.
In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $Δu = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
Optimal Rigidity Results for the $k$-Hessian Equation of Lane--Emden Type
Beihang University（北京航空航天大学）；University of Macau（澳门大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
We verify the existence of radial positive solutions for the semi-linear equation $$ -\,Δu=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a radial smooth potential. If $q$ is super-critical, namely $q>p^*$, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided $V(0)<0$. On the other hand, if $N/(N-2)<q<p^*$, we prove that this Problem has a radial solution behaving like a super-position of {\it …
We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in [13]. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining some classical estimates and a new identity …
We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class $C^1$, and that the set of characteristic points is made of concentric spheres in finite number in $\{\frac 1R \le |x|\le R\}$ for any $R>1$.
In this article we consider the system of equations Δu_{i}=p_{i}(x)f_{i}(u_{1},...,u_{d}) for i=1,...,d on R^{N}, N\geq3 and d\in{1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some conditions on p_{i} and f_{i}. Also, we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Furthermore, a non-radially symmetric solution is obtained by using a lower and upper solution method.
We consider the equation $-Δu= |x|^α|u|^{p-1}u$ for any $α\geq 0$, either in $\mathbb R^2$ or in the unit ball $B$ of $\mathbb R^2$ centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as $p\rightarrow +\infty$ of all the radial solutions to these problems. In particular, we show that there is no uniform a priori bound (in $p$) for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the recently shown fact that positive solutions have uniform a priori bounds for $α=0$ and Dirichlet boundary …
