scholarly journals Unilateral Global Interval Bifurcation for the Hessian Equation and Its Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Wenguo Shen
Keyword(s):  
Global Bifurcation ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Quasilinear Problems

In this paper, we establish a unilateral global bifurcation result from the interval for the k-Hessian equations with nondifferentiable nonlinearity. By applying the above result, we shall prove the existence of the principal half-eigenvalues for the half quasilinear problems. Furthermore, we shall determine the interval of γ, in which there exist one-sign solutions for the following k-Hessian equations: SkD2u=αx−u+k+βx−u−k+γaxfu, in B,ux=0, on ∂B.

scholarly journals Generalised solutions of Hessian equations

1997 ◽  
Vol 56 (3) ◽  
pp. 459-466 ◽  
Author(s):  
Andrea Colesanti ◽  
Paolo Salani
Keyword(s):  
Weak Solutions ◽  
Convex Functions ◽  
Symmetric Function ◽  
Convex Set ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Definition Of ◽  
Monge Ampere

We introduce a definition of generalised solutions of the Hessian equation Sm(D2u) = f in a convex set ω ⊂ ℝn, where Sm(D2u) denotes the m-th symmetric function of the eigenvalues of D2u, f ∈ Lp(ω), p ≥ 1, and m ∈ {1, …, n}. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

scholarly journals On critical exponents of a k-Hessian equation in the whole space

2019 ◽  
Vol 149 (6) ◽  
pp. 1555-1575 ◽  
Author(s):  
Yun Wang ◽  
Yutian Lei
Keyword(s):  
Critical Exponents ◽  
Decay Rates ◽  
Classical Solutions ◽  
Sobolev Type ◽  
Extremal Functions ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Radial Structure ◽  
Whole Space ◽  
Image Position

AbstractIn this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation $$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$with radial structure, where n ⩾ 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.

scholarly journals Complex Hessian Equations on Some Compact Kähler Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-48 ◽  
Author(s):  
Asma Jbilou
Keyword(s):  
Smooth Function ◽  
A Priori ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Continuity Method ◽  
Bisectional Curvature ◽  
Hessian Equation ◽  
Holomorphic Bisectional Curvature ◽  
Hessian Equations ◽  
Kähler Form

On a compact connected2m-dimensional Kähler manifold with Kähler formω, given a smooth functionf:M→ℝand an integer1<k<m, we want to solve uniquely in[ω]the equationω̃k∧ωm-k=efωm, relying on the notion ofk-positivity forω̃∈[ω](the extreme cases are solved:k=mby (Yau in 1978), andk=1trivially). We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

scholarly journals The Dirichlet Problem of Hessian Equation in Exterior Domains

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 666
Author(s):  
Hongfei Li ◽  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Viscosity Solutions ◽  
Exterior Domains ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations det D 2 u = f ( x ) .

Boundary behavior of large solutions to a class of Hessian equations

2020 ◽  
pp. 1-16
Author(s):  
Ling Mi ◽  
Chuan Chen
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Nonlinear Term ◽  
Boundary Behavior ◽  
Singular Boundary ◽  
Structure Condition ◽  
Hessian Equation ◽  
Large Solutions ◽  
Hessian Equations ◽  
Singular Boundary Condition

In this paper, we consider the m-Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω. We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

Sign in / Sign up

Export Citation Format

Share Document