scholarly journals A characterization of semistable radial solutions of k-Hessian equations

2021 ◽  
Vol 497 (2) ◽  
pp. 124902
Author(s):  
Miguel Angel Navarro ◽  
Justino Sánchez
Keyword(s):  
Radial Solutions ◽  
Hessian Equations

Sharp estimates of semistable radial solutions of k-Hessian equations

2019 ◽  
Vol 150 (4) ◽  
pp. 2083-2115 ◽  
Author(s):  
Miguel Angel Navarro ◽  
Justino Sánchez
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Extremal Solution ◽  
Nonincreasing Function ◽  
Pointwise Estimates ◽  
Radial Solutions ◽  
Sharp Estimates ◽  
Dirichlet Data ◽  
Hessian Equations ◽  
Hessian Operator

AbstractWe consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

scholarly journals Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations

2018 ◽  
Vol 23 (4) ◽  
pp. 475-492 ◽  
Author(s):  
Jianxin He ◽  
Xinguang Zhang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Growth Conditions ◽  
Radial Solutions ◽  
Local Growth ◽  
Hessian Equations ◽  
Existence And Nonexistence ◽  
The Fixed Point Theorem

In this paper, we establish the existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations in a ball. Under some suitable local growth conditions for nonlinearity, several new results are obtained by using the fixed-point theorem.

scholarly journals Radial solutions for fully nonlinear elliptic equations of Monge–Ampère type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Limei Dai ◽  
Huihui Cheng ◽  
Hongfei Li
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Classical Solutions ◽  
Radial Solutions ◽  
Fully Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Hessian Equations ◽  
Existence And Nonexistence ◽  
Moving Plane ◽  
Monge Ampere

AbstractFirst, the symmetry of classical solutions to the Monge–Ampère-type equations is obtained by the moving plane method. Then, the existence and nonexistence of radial solutions in a ball are got from the symmetry results. Finally, the existence and nonexistence of classical solutions to Hessian equations in bounded domains are considered.

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