Iterative properties of solution for a general singular -Hessian equation with decreasing nonlinearity☆
Introduction
This paper focuses on the iterative properties of solution for the following general singular -Hessian equation with Dirichlet boundary condition where is a unit ball, is continuous, is a -Hessian operator defined as where are the eigenvalues of the Hessian matrix , is the vector of eigenvalues of . The -Hessian operator is actually a second order fully nonlinear differential operator, in particular, if , it is the standard Monge–Ampére operator [1] and if , it becomes the well known classical Laplace operator [2], [3], [4], [5], [6], [7]. Hence, the -Hessian operators are a discrete collection of partial differential operators including both the Laplace and the Monge–Ampére operator.
In mathematics application, Hessian operator arises in the investigation of some remarkable geometrical problems [8] and quasilinear parabolic problems [9], and have been studied by many authors in different settings, for details, see [10], [11], [12], [13], [14], [15], [16]. Especially, under some general structure, Guan [10] derived some second order priori estimates of solutions for a fully nonlinear elliptic Hessian type equation on Riemannian manifolds, and then the regularity and existence of solutions are studied by using these estimates. In recent work [11], Ji and Bao showed -Hessian equation has a positive subsolution if and only if satisfies Keller–Osserman condition where is nonnegative, increasing and continuous. In [12], He et al. dealt with a class of Dirichlet problems of general -Hessian equations involving a nonlinear operator in a ball, by introducing suitable local growth conditions for nonlinearity and using the fixed point theorem, several new results of the existence, nonexistence and multiplicity of radial solutions were established, which improved and generalized some recent work.
However, comparing with numerous existing results on Hessian equations, it is not difficult to find that there are few work that allow nonlinearity to have singularity at time or space variables, especially, some increasing conditions were required such as [11], [15]. Inspired by the existing work, in this paper we establish the uniqueness of solution and its converges properties for the above singular -Hessian Dirichlet boundary value problem. Here it is noteworthy that the nonlinear function involved in Eq. (1.1) is decreasing and can be allowed a stronger singularity at some points of the time or space variables. Actually, singular behaviour is a class of interesting natural phenomena occurring in many complex physical, biological and mechanics fields, which has been studied extensively by many authors via to various strategies and techniques[17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]. However, if singular behaviour happens at some points or domain of space variables, the handling of singularity is difficult and some complicated conditions are normally required. For example, Zhang et al. [23] used a complicated supremum-limit restrained condition to carry out the singularity analysis for a fractional differential equation with signed measure. On the other hand, to perform an iterative process, most of existing work required the nonlinearity of equation being increasing in space variables and no singularity at space variables, see [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]. If the nonlinearity is a decreasing function, it is more difficult to deal with singularity because it is hard to find a suitable lower solution of the equation, thus the upper and the lower solution method is not an effective strategy [17], [59]. Moreover, without of the help of increasing condition, even for the simplest case, iterative process is hard to perform by using classical iterative techniques.
Thus, in this paper, to overcome the difficulty from singularity and decreasing condition, we introduce a new double iterative technique to explore the uniqueness of solution and its iterative properties. The rest of this paper is organized as follows. In Section 2, some preliminaries and lemmas are recalled for further developments. The main results are presented in Section 3.
Section snippets
Preliminary results on radial solutions
In this paper, we are only interested in the classical solutions of the -Hessian equation (1.1), that is, a radial function satisfies the -Hessian equation (1.1).
Let be radius of balls and with . The following lemma has been proven in [11]:
Lemma 2.1 Let be a radially symmetric function and . Then the function with is , and [11]
Main results
Before stating our main results, we firstly list the following conditions:
is continuous and nonincreasing in the second variable with
there exist a constant and a function with such that for any and ,
Remark 3.1 The assumption implies that the nonlinear term can be singular at and (or) .
Remark 3.2 If satisfies the assumption , then the inequality (3.1) is equivalent
An example
In this section, we present an example to illustrate our main results.
Example 4.1 Consider the following general singular -Hessian Dirichlet value boundary problem where is a unit ball. For the 3-Hessian equation (4.1), we have the following conclusions: the problem (4.1) has a unique classical radial solution in ; for any initial value , the iterative sequence :
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