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

School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China hjx_cheng@163.com School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China zxg123242@163.com School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China mathlls@163.com Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia y.wu@curtin.edu.au


Introduction
In this paper, we consider the existence and nonexistence of radial solutions for the following Dirichlet problem of the general k-Hessian equation: where Ω is a unit ball, f : C(Ω × R → [0, +∞)), B ∈ X is a nonlinear operator with the following property: +∞), [0, +∞) : there exists a constant σ > 0 such that for any 0 < c < 1, B(cs) c σ B(s) .S 1/k k (λ(D 2 u)) is defined as the k-Hessian operator by where λ 1 , λ 2 , . . ., λ N are the eigenvalues of the Hessian matrix D 2 u, and λ(D 2 u) = (λ 1 , λ 2 , . . ., λ N ) is the vector of eigenvalues of D 2 u.Clearly, S k (λ(D 2 u)) is a secondorder fully nonlinear differential operator for k > 1, which is the sum of all k×k principal minors of the Hessian matrix of D 2 u.On the other hand, the k-Hessian operator can also be written in the divergence form where S ij k = ∂S k (D 2 u)/∂u ij , and for more details, the reader is referred to [13,29,30].It is easy to see that the k-Hessian operator is a generalization of both the Monge-Ampère operator [1,2] when k = N and the Laplace operator [15] when k = 1.This implies that the k-Hessian operator constructs a discrete collection of partial differential operators including the Monge-Ampère operator and the Laplace operator as special cases.
In many existing work for the k-Hessian equation, mathematical theories are constructed with no background on modeling or exploration of their applications.We thus briefly review here some potential applications in physics and applied mathematics.In [9,11], Escudero used the k-Hessian equation to model various phenomena of condensed matter and statistical physics.In addition, the k-Hessian equation is also regarded as an important class of fully nonlinear operators related to an object of geometric investigation [31,32] and study of quasilinear parabolic problems [26].
There are many rich literatures concerning the k-Hessian equation.For example, Caffarelli, Nirenberg, and Spruck [3] first studied the existence and a priori estimate of the smooth solutions for the k-Hessian equation Then the work was extended to more general equations in [21,28], and for more recent results, we refer the reader to [4-6, 10, 12, 17, 26, 27, 33].In [20] and [12], the regularity for a more general class of fully nonlinear elliptic equations was obtained under nondivergence form.Recently, Covei [6] considered the existence of positive radial solutions for a Hessian equation with weights and a system of two Hessian equations By using a successive approximation technique, a necessary condition and a sufficient condition for a positive radial solution to be large were established.Inspired by the above work, in this paper, we establish the existence and nonexistence of radial solutions for the k-Hessian equation (1) based on some fixed point theorems.Noticing that the k-Hessian equation ( 1) involves a nonlinear operator B, so it includes many interesting and important cases.In particular, if B(x) = x k−1 , then the k-Hessian equation ( 1) reduces to the Hessian equation ( 2), which has been studied by many authors [3,4,14,21,28] via to different methods such as the variational method, the Perron's method, and so on.Moreover, if B(x) = const = 0, then the k-Hessian equation becomes Hessian equation (3).Ji and Bao [17], Covei [6] considered the necessary and sufficient conditions for the existence of positive radial solutions.When B(x) = |x| p−2 , p 2, the k-Hessian equation (1) becomes which is a p-Poisson-Hessian equation, and few work were reported.Thus, the k-Hessian equation ( 1) is a generalization of fully nonlinear elliptic equations involving many important cases.To the best of our knowledge, no results have been reported on the existence and nonexistence of radial solutions for the k-Hessian equation (1), and this is the first paper using the Leggett-Williams' fixed point theorem and the Leray-Schauder nonlinear alternative theorem to study the k-Hessian equation involving a nonlinear operator.Before we give a detailed description of our main results, we first establish the following property of the inverse operator of the operator sB(s).Proposition 1.If B ∈ X , let L(s) = sB(s), then L has a nonnegative increasing inverse mapping L −1 (s), and for any 0 < b < 1, Proof.Firstly, we prove that B is an increasing operator if B ∈ X .In fact, for any B ∈ X and s, t ∈ [0, +∞), without loss of the generality, let 0 s < t.If s = 0, obviously B(s) B(t) holds.If s = 0, let c 0 = s/t, then 0 < c 0 < 1.It follows from the property of B that B(s) = B(c 0 t) c σ 0 B(t) B(t), which implies that B is an increasing operator.Thus, we have L (s) = (sB(s)) > 0 for any s > 0, which implies that L is a bijection on (0, ∞) and has a nonnegative increasing inverse mapping L −1 (s).
Remark 1.Clearly, if r 1, then we have Remark 2. The operator set X includes a large class of operators and the standard type of operators is B(s) = n i=1 s αi , α i > 0. In fact, take σ = min{α 1 , . . ., α n } > 0, then for any 0 < c < 1, one has B(cs) c σ B(s).

Preliminary results on radial solutions
In this paper, we only focus on the classical solutions of the k-Hessian equation ( 1), namely, a function u(t) of class C 2 [0, 1] satisfies the k-Hessian equation (1).In the rest of this paper, t is used as an independent variable of functions, and r as radiuses of balls in the cone.For B R := {x ∈ R N : |x| < R} and radial function u(r) with r = N i=1 x 2 i , we have the following properties.
https://www.mii.vu.lt/NA then we have Thus, from Lemma 1 and Proposition 1 we get the following lemma.
Now with a simple transformation ϕ = −v, (5) can be rewritten as follows: Define the Banach space E = C[0, 1] with the usual supremum normal ϕ(x) = max x∈[0,1] |ϕ(x)|, and define a nonlinear operator F on E as follows: We will establish conditions for the existence, nonexistence, and multiplicity of radial solutions for equation (1) in Sections 3-5, respectively.

Existence results
Nonlinear functional analysis method plays an important role for studying nonlinear ordinary differential equations and partial differential equations [7, 16, 18, 19, 23-25, 34-37, 40-49].Many fixed point theorems have been developed to solve various boundary value problems of differential equations [7,38,39].In this section, our main tool to establish a existence result of solution for k-Hessian equation ( 1) is the following Leray-Schauder nonlinear alternative theorem [8].Lemma 3. Let E be a real Banach space, Ω be a bounded open subset of E, 0 ∈ Ω, L : Ω → E is a completely continuous operator.Then either there exist ϕ ∈ ∂Ω and µ > 1 such that L(ϕ) = µϕ or there exists a fixed point ϕ * ∈ Ω.
Theorem 1. Assume that there exist a nondecreasing function ψ : [0, +∞) → [0, +∞) and a function a(t Then the k-Hessian equation (1) has at least one solution if there exists a real number m > 0 such that Proof.Firstly, we prove that the operator F is completely continuous.Clearly, continuity of the operator F follows from the continuity of f .Let D ⊂ E be any bounded set.Then there exists a constant L > 0 such that |f (r, ϕ)| L for any (r, ϕ) ∈ [0, 1] × D, thus, we have , and then, for any ϕ ∈ D and r 1 , r 2 ∈ [0, 1], we get It follows from (8) that F(E) is equicontinuous on [0, 1].Thus, according to the Ascoli-Arzela theorem, F is a completely continuous operator.Now we consider B m = {ϕ ∈ C[0, 1]: ϕ m}.It follows from the Leray-Schauder nonlinear alternative theorem that either the operator F has a fixed point or there exists ϕ ∈ ∂B m such that Fϕ = µϕ for some µ > 1.We assert that the latter conclusion does not hold.Otherwise, there exist some ϕ 0 ∈ ∂B m and some µ > 1 such that Fϕ 0 = µϕ 0 .Thus, it follows from ( 6)-( 7) that that is µ 1, which leads to a contraction with µ > 1.In consequence, the operator F has a fixed point in C[0, 1] with ϕ m.This further implies that problem (1) has at least one solution on [0, 1] if (7) holds.The proof is completed.
By Theorem 1 we have the following corollary.
Corollary 1. Assume that there exists a function a(t) ∈ C[0, 1] such that Then the k-Hessian equation (1) has at least one solution.

Nonexistence results
In this section, we are interested in the nonexistence result of solutions for the k-Hessian equation ( 1) with a parameter µ: Let then we have the following nonexistence result of solutions.

Results on multiple solutions
In order to obtain the multiplicity of radial solutions of (1), we need the following Leggett-Williams fixed point theorem.
Definition 1.Let P be a cone in a real Banach space E. A mapping α is called a nonnegative continuous concave functional on P if it satisfies (i) https://www.mii.vu.lt/NASuppose that α is a nonnegative continuous concave functional on P , for constants 0 < a < b and c > 0, define the following convex sets: Then F has at least three fixed points u 1 , u 2 , u 3 satisfying

Now let
then P is a cone in E. We still consider the nonlinear operator F on E: For any ϕ ∈ P , clearly, (Fϕ)(r) 0 for all r ∈ [0, 1], (Fϕ) (0) = (Fϕ)(1) = 0, and Thus, we have F : P → P .On the other hand, by the standard argument we know that F is continuous and compact, also see [43].So, from the above facts we have the following lemma.
Lemma 5. F : P → P is continuous and compact.Now for some µ 0 ∈ (0, 1/2), define a nonnegative continuous concave functional In what follows, we define two constants: , where σ is defined by B, which depends on the operator B. Clearly, it follows from N > 2k that λ 1 > 1.
Theorem 3. Assume that there exist four constants a, b, c, d such that 0 < d < a < µ 0 (1 − µ 0 )b < b c, and the following conditions are satisfied: Then the k-Hessian equation (1) has at least three radial solutions satisfying Proof.Firstly, from the definition of α we have α(ϕ) ϕ for ϕ ∈ P .Now we prove that F : P c → P c , and for any ϕ ∈ P d , there is Fϕ d.In fact, for any ϕ ∈ P c , we have ϕ c, and it follows from (A3) and ( 4) that which implies that F : P c → P c .In the same way, we have Fϕ < d for any ϕ ∈ P d .Thus, condition (ii) of Lemma 4 is satisfied.
Thirdly, we verify that condition (iii) of Lemma 4 is satisfied.For any ϕ ∈ P (α, a, c) with Fϕ > b, we have Then we have and On the other hand, for ϕ ∈ P (α, a, c) with Fϕ > b, by ( 10) and (A4), we have Thus, all hypotheses of the Leggett-Williams theorem are satisfied.So, according to the Leggett-Williams theorem, the k-Hessian equation (1) has at least three radial solutions satisfying Corollary 2. Assume that there exist three constants a, b, d such that 0 < d < a < µ 0 (1 − µ 0 )b and (A1), (A2), and the following condition are satisfied: Then the k-Hessian equation (1) has at least three radial solutions satisfying Corollary 3. Suppose that (A2) and (A4) hold.In addition, assume the following conditions are satisfied: Then the k-Hessian equation (1) has at least three radial solutions satisfying Proof.In fact, clearly, (A5) implies (A1).So, we only need to prove that there exists a positive constant c with c b such that F : P c → P c .It follows from (A6) that there exists a constant δ > 0 such that f (r, s) λ 1 L(s) ∀r ∈ [0, 1], s δ.
Proof.In fact, here k = 2 and f (t, u) = t 1/2 sin 2 u.Since B(x) = x 2 , we have L(x) = x 3 and and by Theorem 1, the 2-Hessian equation ( 11) has at least one solution.
Example 2. Consider the nonexistence of radial solutions for the following Dirichlet problem of the 2-Hessian equation with a parameter where Ω is a unit ball.Then there exists µ 0 > 0 such that the 2-Hessian equation ( 12) has no solutions for 0 < µ < µ 0 .
Example 3. Consider the existence of multiple radial solutions for the following Dirichlet problem of the 2-Hessian equation: where Ω is a unit ball and ( Then the 2-Hessian equation ( 14) has at least three solutions satisfying  Therefore, condition (A2) also holds.

Conclusion
The k-Hessian equation is a class of very important fully nonlinear and nonuniformly elliptic partial differential equations, which fill up the gap between the Monge-Ampère and Poisson equations.In this paper, we introduce a nonlinear operator B such that k-Hessian equation we studied include many important and interesting cases.To establish the existence, nonexistence, and multiplicity of radial solutions to Dirichlet problems of k-Hessian equations with a nonlinear operator in a ball, we adopt the Leray-Schauder alternative theorem and the Leggett-Williams fixed point theorem as well as some suitable growth conditions for nonlinearity.Our work improves and generalizes some recent work such as [3,4,6,17,21,28].