Unilateral Global Interval Bifurcation for the Hessian Equation and Its Applications

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 c, in which there exist one-sign solutions for the following k-Hessian equations: Sk(D 2u) � α(x)(− u+) + β(x)(− u ) + ca(x)f(u), inB, u(x) � 0, on ∂B. 􏼨


Introduction
We first consider the following k-Hessian equation: where D 2 u � (∂ 2 u/∂x i ∂x j ) is the Hessian matrix of u, B is the unit ball of R N , a(x) is a weighted function, λ is a positive parameter, and g ∈ C(B × R 2 ). Here, S k (D 2 u) � σ k (λ(D 2 u)) � 1≤i 1 < ···<i k ≤ N λ i 1 . . . λ i k is the k-Hessian operator of u, where λ(D 2 u) ≔ λ � (λ 1 , . . . , λ k ) is the eigenvalues of D 2 u.
e N-Hessian equation is just the Monge-Ampère equation.
Recently, in [15], the radial solutions of problem (1) is equivalent to the solutions of the following boundary value problem: where r � |x| with x ∈ B. By a solution of problem (2), we understand that it is a function which belongs to C 2 [0, 1] and satisfies (2). Dai [15] established a global bifurcation result for the k-Hessian equation (2) with λa(r)(− u) k + g(r, − u, λ) equal λ k ((− u) k + g(− u)).
However, in [15], the nonlinearities are differentiable at the origin. In [16], Berestycki established an important global bifurcation theorem from intervals for a class of second-order problems involving nondifferentiable nonlinearity. Recently, Dai and Ma [17,18] considered one-dimensional and high-dimensional p-Laplacian problems involving nondifferentiable nonlinearity, respectively. In 2018, Shen [19,20] established interval bifurcation theorems for the Kirchhoff-type problems and the Monge-Ampère equations involving nondifferentiable nonlinearity, respectively. Motivated by the above papers, in this paper, we shall establish a unilateral global interval bifurcation result for the following k-Hessian equations with nondifferentiable nonlinearity: where B is the unit open ball of R N and the nonlinear term F has the form F � f + g, where f, g ∈ C(B × R 2 ) are radially symmetric with respect to x. It is clear that the radial solutions of (3) is equivalent to the solutions of the following problem: On the other hand, half-linear or half-quasilinear problems have attracted the attention of some specialists (see [16][17][18][19][20]). On the above basis of the unilateral global interval bifurcation result, we shall study the following half-quasilinear eigenvalue problem: It is clear that the radial solutions of (5) is equivalent to the solutions of the following problem: Following the above eigenvalue theory (see eorem 2), we shall investigate the existence of radial one-sign solutions for the following problem: It is clear that the radial solutions of (7) is equivalent to the solutions of the following problem: e rest of this paper is arranged as follows. In Section 2, we give some preliminaries. In Section 3, we establish the unilateral global interval bifurcation result for k-Hessian equation (4) (see eorem 1). In Section 4, on the basis of the unilateral global interval bifurcation result, we shall prove the existence of the principal eigenvalues for k-Hessian equation (6) (see eorem 2). In Section 5, we shall investigate the existence of one-sign solutions for k-Hessian equation (8).
By Sections 5 and 6 in [15], we firstly consider the following problem: Let us define the operator G k (h): E ⟶ E by For a given h ∈ Y, G k (h) : Y ⟶ E is a completely continuous and (10) is equivalent to (9).
With a simple transformation v � − u, problem (2) can be equivalently written as where g ∈ C([0, 1] × (R + ) 2 ) and a satisfies the following conditions: and λ on bounded sets And g also satisfies the following subcritical growth restriction (see [15], p.4-(1.5)): uniformly for r ∈ (0, 1), λ on bounded sets, and some q ∈ (0, k * ). Here, is the critical exponent for the k-Hessian operator, which was first determined in [21]. Define the Nemytskii operator H: en, it is clear that H is a continuous (compact) operator and problem (11) can be equivalently written as where F : R × E ⟶ E is a completely continuous and F(μ, 0) � 0, ∀μ ∈ R. Now, we consider the following eigenvalue problem: By the same proof as in eorem 4.1 of [14], we can show that problem (16) possesses the first eigenvalue λ 1 which is positive, simple, unique and the corresponding eigenfunctions are positive in (0, 1).
By Rabinowitz [22], using the same method to prove ( [15], eorem 1.1) with obvious changes, we may get the following global bifurcation result.
Next, we give the following results.

Lemma 3. Let I be an interval and if y and z are functions
Discrete Dynamics in Nature and Society . en, we have the following identity: Proof. Similar the proof of Lemma 2.3 of [15] with obvious change, the left-hand side of (18) equals d dt □ Lemma 4. In (18) of Lemma 3, by Young's inequality, we get and the equality holds if and only if sgny � sgnz and |y ′ /y| k+1 � |z ′ /z| k+1 .
By Lemma 3 and Lemma 4, we have the following result: □ Lemma 5. In (18) of Lemma 3, we have Proof. By (18) of Lemma 3, it follows that Similar to the proof as (2.3) in Lemma 2.3 of [15], we can show that the left-hand side of (23) equals 0. By Lemma 4, we have the result.
In order to treat the problems with nonasymptotic nonlinearity at 0 and ∞, we use Whyburn-type superior limit theorems. However, if the collection of the infinite sequence of sets is unbounded, Whyburn's limit theorem ( [24], eorem 9.1) cannot be used directly because the collection may not be relatively compact. In order to overcome this difficulty, Dai [25] established the following results. □ Lemma 6 (see [25], Lemma 2.5). Let X be a normal space and let C n | n � 1, 2, . . . be a sequence of unbounded connected subsets of X. Assume that (i) ere exists z * ∈ lim inf n⟶+∞ C n with ‖z * ‖ < +∞.

4
Discrete Dynamics in Nature and Society en, D ≔ lim sup n⟶∞ C n is unbounded, closed, and connected.
Lemma 7 (see [25], eorem 1.2). Let X be a normal vector space and let C n | n � 1, 2, . . . be a sequence of unbounded connected subsets of X. Assume that en, D ≔ lim sup n⟶∞ C n is unbounded, closed, and connected.
Lemma 8 (see [25], Corollary 2.3). Let (X, ρ) be a metric space. If C i i∈N is a sequence of sets whose limit superior is L and there exists a homeomorphism T: X ⟶ X such that for every R > 0, ( ∪ +∞ i�1 T(C i )) ∩ B R is a relatively compact set; then, for each ϵ > 0, there exists an m such that for every n > m, C n ⊂ V ϵ (L), where V ϵ (L) denotes the set of all points p with ρ(p, x) < ϵ for any x ∈ L.

Global Bifurcation from an Interval
With a simple transformation v � − u, problem (4) can be equivalently written as , g satisfying (A1) and (12), and f satisfying Let S ] denote the closure in R × E of the set of nontrivial solutions (λ, v) of (26) with v ∈ P ] . By an argument similar to that of ( [15], Lemma 5.1) with obvious changes, we can show that the following existence and uniqueness theorem is valid for problem (26).
Lemma 10. Let ϵ n , 0 < ϵ n < 1, be a sequence converging to 0. If there exists a sequence (λ n , v n ) ∈ K ] such that it is a nontrivial solution of problem (27) corresponding to ϵ � ϵ n and (λ n , v n ) converges to (λ, 0) in R × E, then λ ∈ I.
Proof. Let w n � v n /‖v n ‖, then w n satisfies the following problem: Discrete Dynamics in Nature and Society Let g(r, v, λ) � max 0≤|s|≤v |g(r, s, λ)|, for all r ∈ (0, 1) and λ on bounded sets, then, g is nondecreasing with respect to v and uniformly for r ∈ (0, 1) and λ on bounded sets. Furthermore, it follows from (30) that uniformly for r ∈ (0, 1) and λ on bounded sets. Clearly, (A2) implies that for all r ∈ (0, 1). Note that ‖w n ‖ � 1 implies ‖w n ‖ ∞ ≤ 1. Using this fact with (31) and (32), we have that λ n Nr N− 1 a(r)w k n + Nr N− 1 f(r, v n |v n | ϵ n , λ n )/‖v n ‖ k + Nr N− 1 g(r, v n , λ n )/‖v n ‖ k is bounded in E for n large enough. e compactness of G k implies that w n is converges in E. Without loss of generality, we may assume that w n ⟶ w in E with ‖w‖ � 1. Clearly, we have w ∈ P ] .
We claim that w ∈ P ] . On the contrary, suppose that w ∈ ∂P ] , by Lemma 9, then, w ≡ 0, which is a contradiction with ‖w‖ � 1. Now, we deduce the boundedness of λ. Let ψ be an eigenfunction of problem (16) corresponding to λ 1 .

□
Proof of eorem 1. We only prove the case of ] � + since the proof of the case ] � − can be given similarly. Let C + be the component of S + ∪ (I × 0 { }). First, using the similar method to prove ( [17], eorem 1.1) with obvious changes, we may prove that C + ⊂ (K + ∪ (I × 0 { })).
In the following, we prove that C + is unbounded. Suppose on the contrary that C + is bounded. Using the similar method to prove eorem 1 in [16] with obvious changes, we can find a neighborhood O of C + such that ∂O ∩ P + � ∅.
Since o is bounded in K + , equation (27) shows that (λ ϵ , v ϵ ) is bounded in R × C 2 independently of ϵ. By the compactness of G k , one can find a sequence ϵ n ⟶ 0 such that (λ ϵ n , v ϵ n ) converges to a solution (λ, v) of (27). So v ∈ P + . If v ∈ ∂P + , then from Lemma 9, it follows that v ≡ 0. By Lemma 10, λ ∈ I, which contradicts the definition of O. On the other hand, if v ∈ P + , then (λ, v) ∈ P + ∩ ∂O which contradicts P + ∩ ∂O � ∅.
From eorem 1 and its proof, we can easily get the following corollaries. We relax the assumption of a(x) as the following.

Spectrum of Half-Quasilinear Eigenvalue Problems
With a simple transformation v � − u, problem (7) can be equivalently written as Discrete Dynamics in Nature and Society { }, and α(r) and β(r) satisfy the following: (A3) α (r), β (r) ∈ C[0, 1] are radially symmetric. Problem (40) is called half quasi-linear because it is positive k homogeneous in the cones u > 0 and u < 0. Similar to that of [16], we say that λ is a half-eigenvalue of problem (40), if there exists a nontrivial solution (λ, u λ ). λ is also said to be simple if v � cu λ , c > 0 for all solutions (λ, v) of problem (40). A half-eigenvalue is called the principal halfeigenvalue if the corresponding eigenfunction is positive or negative.
By an argument similar to that of [17,18], we can obtain the following theorem.
Proof. Applying the similar method to prove eorem 1.2 in [17] or eorem 3.1 in [18] with obvious changes, by eorem 1 and Lemma 3-5, we can prove the conclusion.
Naturally, we can consider the bifurcation structure of the perturbation of problem (40) of the following form: where g satisfies (A1) and (12).
□ Remark 1. By some simple computation, we can show that if Remark 2. Note that min λ + , λ − is the least half-eigenvalue for (40). Using a similar method to prove eorem 3.1 in [17] or eorem 3.2 in [18] with obvious changes, by Corollary 1, we may obtain the following result.

One-Sign Solutions for the Hessian Equation
With a simple transformation v � − u, problem (8) can be equivalently written as where f satisfies the following subcritical growth restriction (see [15], (6.1)): for some p ∈ (k, k * ]. Here, ese are the lower critical exponent for the k-Hessian operator (see [26,27]).
We assume that f satisfies the following assumptions: Discrete Dynamics in Nature and Society Here, Obviously, any one-sign solution of problem (42) is a one-sign solution of problem (8). Conversely, we claim that any one-sign solution of problem (8) is a one-sign solution of problem (42).
Define the Nemytskii operator en, it is clear that H 1 is a continuous (compact) operator and problem (42) can be equivalently written as where F 1 : R × E ⟶ E is completely continuous and e main results of this section are the following theorems.  Remark 5. Note that if α(t) � β(t) ≡ 0, the result of eorem 4 is equivalent to those of eorem 6.3 in [15]; eorem 4 extends the corresponding results of [15]. In order to prove eorem 4, similar to the proof of Lemma 2.3 of [15] and Lemma 3.1 of [17] with obvious change, we need the following Sturm-type comparison result.
Also, let v 1 and v 2 be solutions of the following differential equations: Proof. If u 2 (r) � 0 on (c, d), we discuss two cases.
By Lemma 11, we obtain the following result that will be used later.
Step 1. We show that there exists a constant number M > 0 such that μ n ∈ (0, M], for all n. On the contrary, we suppose that lim n⟶+∞ |μ n | � +∞. Since (μ n , v n ) ∈ D ] , it follows from the compactness of G k that Discrete Dynamics in Nature and Society By Remark 4, we have Let ψ ] be an eigenfunction corresponding to λ ] . But if lim n⟶+∞ μ n f n (r) � − ∞, applying Lemma 12 to v n and ψ ] , we have that ψ ] must change sign for n large enough, by ψ ] ∈ P ] , which is impossible. So lim n⟶+∞ μ n f n (r) � +∞, applying Lemma 12 to ψ ] and v n , we get that v n must change sign for n large enough, and this contradicts the fact that v n ∈ P ] .
Step 2. We show that D ] joins (λ ] /f 0 , 0) to (λ ] /f ∞ , ∞). It follows from Step 1 that ‖v n ‖ ⟶ + ∞. Let It follows that We divide the equation by ‖v n ‖ k and set w n � v n /‖v n ‖. Since w n is bounded in C 2 [0, 1], choosing a subsequence and relabeling if necessary, we have that w n ⟶ w for some w ∈ E. Moreover, from (54) and the fact that ξ is nondecreasing, we have that n ⟶ ∞, ∀r ∈ [0, 1].

(57)
By the compactness of G k , we obtain that r N− k − w n ′ k ′ � N C k N r N− 1 μ n a(r)f ∞ w n + α(r) w + n k + β(r) w − n k , r ∈ (0, 1), w n ′ (0) � w n (1) � 0, where μ ≔ lim n⟶∞ μ n again choosing a subsequence and relabeling if necessary. us, it is clear that ‖w‖ � 1 and w ∈ D ] ⊆ D ] since D ] is closed in R × E. Moreover, by eorem 2, μf ∞ � λ ] , so that Proof. We shall only prove the case of λ ∈ (0, λ 2 ) since the proofs of the other cases are completely analogous. Clearly, f 0 , f ∞ ∈ (0, +∞) implies that there exists a positive constant M such that f(s) s k ≤ M, for any s ≠ 0.
It is sufficient to show that there exists λ 2 > 0 such that problem (42) has no solution for all λ ∈ (0, λ 2 ). Suppose on the contrary that there exists one pair (μ, v) ∈ D + such that μ ∈ (0, ((C k N − α 0 − β 0 )/(ca 0 M))). Let w � v/‖v‖. Obviously, one has that is is a contradiction. From the Proof of eorem 4 and Remark 6, we can deduce the following two corollaries.