# New Results on the Radial Solutions to a Class of Nonlineark-Hessian System

Authors: Guotao Wang; Zhuobin Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper investigates the positive radial solutions of a nonlineark-Hessian system. ΛSk1/kλD2z1Sk1/kλD2z1=bxφz1,z2,x∈ℝNΛSk1/kλD2z2Sk1/kλD2z2=hxψz1,z2,x∈ℝN, where Λ is a nonlinear operator and b, h, φ, ψ are continuous functions. With the help of Keller–Osserman type conditions and monotone iterative technique, the positive radial solutions of the above problem are given in cases of finite, infinite, and semifinite. Our results complement the work in by Wang, Yang, Zhang, and Baleanu (Radial solutions of a nonlinear k-Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91(2020), 105396).


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## Body

## 1. Introduction

It is well known that Laplace equation has a wide range of applications in mathematics and physics, for instance [1], it can be used to describe the relationship between the curvature of liquid surface and the surface pressure of liquid [2] as well as in solving electrostatic field problems [3, 4]. In a mathematical sense, the existence of solutions of Laplace equation has attracted increasing attention, and numerous excellent results have been obtained.In 1957, under the condition(1)∫α∞1∫0tφsdsdt<∞,∀α>0.Keller [5] studied the existence of solutions for the nonlinear equation Δz=φz, and Osserman [6] studied the existence of solutions for the nonlinear differential inequality Δz≥φz.In 2011, under the Keller–Osserman condition(2)∫1∞1∫0tφsdsdt<∞,∫1∞1∫0tψsdsdt<∞.Peterson and Wood [7] presented the existence of entire positive blow up radial solutions of semilinear elliptic system(3)Δz1=bxφz2,x∈ℝN,Δz2=hxψz1,x∈ℝN.In 2019, under the Keller–Osserman type conditions(4)∫α∞1∫0tφs,ψs,sdsdt<∞,∫β∞1∫0tψφs,s,sdsdt<∞,∀α,β>0.Covei [8] studied the existence and asymptotic behavior of positive radial solutions for the semilinear elliptic system(5)Δz1=bxφz1,z2,x∈ℝN,Δz2=hxψz1,z2,x∈ℝN,where b, h, φ, ψ are continuous functions. Besides, under the Keller–Osserman condition and its extension, many excellent results have been obtained, such as the existence and uniqueness of the nonnegative viscosity solutions [9], the existence and uniqueness of blow up solutions [10], the existence and uniqueness of entire blow up solutions [11], the existence of entire classical weak solutions of the differential inequality [12], the existence and uniqueness of solutions with strong isolated singularity [13], and the existence of solutions for the boundary blow up problem in one dimensional case [14].we define thek-Hessian operator k=1,2,…,N as follows:(6)SkλD2z=∑1≤j1≤…≤jk≤Nλj1⋯λjk,where D2z=∂2zx/∂xi∂xjN×N,i,j=1,2,…,N is the Hessian matrix of z∈C2ℝN and the λD2z=λ1,…,λN∈ℝN is the vector, which consists of eigenvalues of D2z. It is easy to observe that SkλD2z is a family of operators including many well-known operators. For example, for k=1, S1λD2z is Laplace operator, which is studied widely in [15–19], for k=N, SNλD2z is Monge-Ampère operator, which is studied extensively in [20–26].Thek-Hessian equations play an important role in differential geometry [27, 28]. They can describe Weingarten curvature or reflector shape [29] and some phenomena of non-equilibrium phase transitions and statistical physics [30, 31]. In 2019, Zhang and Feng [32] considered the existence and asymptotic behavior of k-convex solution to the boundary blow up problem for the following k-Hessian equation:(7)SkλD2z=bxφz,inΩ,u=+∞,on∂Ω,where b and φ are smooth positive functions and Ω is a smooth, bounded, strictly, convex domain of ℝN with N≥2.In 2020, by means of monotone iterative technique, Wang, Yang, Zhang, and Baleanu [1] established the existence of the entire positive bounded radial solutions and entire positive blow up radial solutions for the following k-Hessian system:(8)ΛSk1/kλD2z1Sk1/kλD2z1=bxφz1,z2,x∈ℝN,ΛSk1/kλD2z2Sk1/kλD2z2=hxψz1,z2,x∈ℝN,where b,h∈C0,∞,0,∞, Λ is a nonlinear operator in the set(9)Θ=Λ∈C20,∞,0,∞|Λls≤lpΛsfor alll∈0,1,s∈0,∞and somep∈0,∞.What interests us in this paper is whether the similar results in [1] hold under the Keller–Osserman type conditions (4) of the k-Hessian system (8). Motivated by the idea, we reconsider the k-Hessian system (8) under the Keller–Osserman type conditions:(10)∫α∞1∫0μφt,ψt,t1/p+1+1k/p+1dtk+1dμ<∞,(11)∫β∞1∫0μψφt,t1/p+1,t+1k/p+1dtk+1dμ<∞.As a continuation of previous work, in this paper, by employing monotone iterative method, we establish some new existence results on positive radial solutions of thek-Hessian system (8) under the cases of finite, infinite, and semifinite. For details of the monotone iterative method, see [1, 7, 8, 33–36]. In addition, our results complement the work in [1] and extend works of many authors in [7, 8, 15–17, 37, 38].


## 2. Preliminaries

For the convenience of subsequent proofs, we list a definition, notations, assumptions, and related lemmas.We first recall the classification of solutions.Definition 1 (see [8]).
A solutionz1,z2∈C20,∞×C20,∞ of system (8) is called an entire bounded solution if condition (12) holds; it is called an entire blow up solution if condition (13) holds; it is called a semifinite entire blow up solution if condition (14) or (15) holds.
Finite case: both componentsz1, z2 are bounded, namely(12)limx⟶∞z1x<∞,limx⟶∞z2x<∞.
Infinite case: both componentsz1, z2 are blow up, namely(13)lim|x|⟶∞z1x=∞,lim|x|⟶∞z2x=∞.
Semifinite case: one of the components is bounded, whereas the other is blow up, namely(14)lim|x|⟶∞z1x<∞,lim|x|⟶∞z2x=∞.
or(15)lim|x|⟶∞z1x=∞,lim|x|⟶∞z2x<∞.
Next, we introduce the notations as follows:r=x and α,β,c1,c2∈0,∞ are suitably chosen,(16)G1r=∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bskds1/kdt,G2r=∫0rktN−k∫0tsN−1CN−1k−1ℜ−1hskds1/kdt,(17)Hr=∫α+βr1φ+ψt,t+1dt,H∞=limr⟶∞Hr,Gr=G1r+G2r,Gi∞=limr⟶∞Gir,i=1,2,(18)ω1r=φ1,βψα,α1/p+1+c2ψ1,1+1αH−1Gr+1ψα,α1/p+1G2r,ω2r=ψαφβ,β1/p+1+c1φ1+1βH−1Gr,1+1φβ,β1/p+1G1r,1,P1r=∫0rkνN−k∫0νtN−1CN−1k−1ℜ−1btφα,β+1ψα,β+11/p+1G2tkdt1/kdν,Q1r=∫0rkνN−k∫0νtN−1CN−1k−1ℜ−1htψα+1φα,β+11/p+1G1t,βkdt1/kdν,(19)ϕ1s=max0≤s≤rbs,ϕ2s=max0≤s≤rhs,(20)P2r=∫0rc1ω1s+1kp+1k+1k+1CN−1k−1sk−1ℜ−1ϕ1skds,k+1Q2r=∫0rc2ω2s+1kp+1k+1k+1CN−1k−1sk−1ℜ−1ϕ2skds,k+1(21)Pi∞=limr⟶∞Pir,Qi∞=limr⟶∞Qir,fori=1,2,(22)F1r=∫αr1∫0μφt,ψt,t1/p+1+1k/p+1dtk+1dμ,F1∞=limr→∞F1r,F2r=∫βr1∫0μψφt,t1/p+1,t+1k/p+1dtk+1dμ,F2∞=limr→∞F2r.
We see that(23)H′r=1φ+ψr,r+1>0,∀r>0,(24)F1′r=1∫0rφt,ψt,t1/p+1+1k/p+1dtk+1>0,∀r>0,(25)F2′r=1∫0rψφt,t1/p+1,t+1k/p+1dtk+1>0,∀r>0,
which mean thatHr has the inverse function H−1r on 0,H∞, F1r has the inverse function F1−1r on 0,F1∞, and F2r has the inverse function F2−1r on 0,F2∞.
We assume thatφ and ψ satisfy the following assumptions:
N1φ,ψ∈C0,∞×0,∞,0,∞ are increasing for each variable and φx,y>0,ψx,y>0 for all x,y>0;
N2 For fixed parameters α,β∈0,∞, there exist c1,c2∈0,∞ such that(26)φt1s1,t2s2≤c1φt1,t2φs1,s2,(27)ψt3s3,t1s1≤c2ψt3,t1ψs3,s1,(28)φα,β≥5−12,ψα,β≥5−12,where t1≥minα,β, s1≥1, t2≥minβ,ψα,α1/p+1, s2≥min1,βψα,α−1/p+1, t3≥minα,φβ,β1/p+1, s3≥min1,αφβ,β−1/p+1; S1P2∞<F1∞<∞,Q2∞<F2∞<∞; S2P1∞<∞,Q1∞<∞; S3F1∞=F2∞=∞,P2∞=Q2∞=∞; S4P1∞=Q1∞=∞; S5F1∞=∞,P1∞=∞,P2∞=∞; S6Q1∞<∞,Q2∞<F2∞<∞; S7F2∞=∞,Q1∞=∞,Q2∞=∞; S8P1∞<∞,P2∞<F1∞<∞.Lemma 1 (see [39]).
IfΛ∈Θ, letting ℜs=sΛs, then
(1):ℜs has a nonnegative increasing inverse mapping ℜ−1s;
(2): when0<l<1, one has(29)ℜ−1ls≥l1/p+1ℜ−1s.
(3): whenl≥1, one has(30)ℜ−1ls≤l1/p+1ℜ−1s.Lemma 2 (see [40]).
We assume thatyr∈C20,R is radially symmetric and y′0=0, then the function zx=yr with r=x<R belongs to C2BR, and(31)λD2z=y″r,y′rr,…,y′rr,r∈0,R,y″0,y″0,…,y″0,r=0,(32)SkλD2z=CN−1k−1y″ry′rrk−1+CN−1ky′rrk,r∈0,R,CNky″0k,r=0,where BR=x∈ℝN:x<R and CNk=N!/k!N−k!.Lemma 3 (see [1]).
z1x,z2x is a radial solution of the k-Hessian system (4) if and only if y1r,y2r is a solution of the following ordinary differential system:(33)rN−kky1r′k′=rN−1CN−1k−1ℜ−1brφy1r,y2rk,r≥0,rN−kky2r′k′=rN−1CN−1k−1ℜ−1hrψy1r,y2rk,r≥0.


## 3. Entire Positive Bounded Radial Solution

In this section, we investigate the entire positive bounded radial solution of system (8), and the main results are as follows.Theorem 1.
We assume thatN1,N2 hold, then system (8) has an entire positive bounded radial solution z1,z2∈C20,∞×C20,∞.Proof.
Obviously, the solutions of system (33) are equivalent to the solutions of the following system:(34)y1r=y10+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bsφy1s,y2skds1/kdt,y2r=y20+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1hsψy1s,y2skds1/kdt.
We define the sequencesy1mrm≥0 and y2mrm≥0 on 0,∞ by(35)y10r=y10=α,y20r=y20=β,r≥0,y1mr=α+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bsφy1m−1s,y2m−1skds1/kdt,r≥0,y2mr=β+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1hsψy1m−1s,y2m−1skds1/kdt,r≥0.
Using the same arguments as in [1], we get the sequences y1mrm≥0 and y2mrm≥0 that are increasing for ∀m≥0 and(36)y1mr+y2mr′φ+ψy1mr+y2mr,y1mr+y2mr+1≤G1′r+G2′r.
Next, integrating the above inequality from 0 tor, we get(37)∫α+βy1mr+y2mrdtφ+ψt,t+1≤Gr.
Consequently,(38)Hy1mr+y2mr≤Gr.
It follows from the above inequality and the fact thatH is a bijection with the inverse function H−1 strictly increasing on 0,H∞ that(39)y1mr+y2mr≤H−1Gr.
Sinceφ, ψ, y1mrm≥0, and y2mrm≥0 are increasing, by means of Lemma 1, N2, (35), (39), we get(40)y1mr≤α+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bsφy1ms,y2mskds1/kdt≤α+φy1mr,y2mr+11/p+1G1r≤α+φy2mr+H−1Gr,y2mr+11/p+1G1r=α+φy2mr1+1y2mrH−1Gr,y2mr+11/p+1G1r≤α+φy2mr1+1βH−1Gr,y2mr+11/p+1G1r≤α+c1φy2mr,y2mrφ1+1βH−1Gr,1+11/p+1G1r=φy2mr,y2mr1/p+1αφy2mr,y2mr1/p+1+c1φ1+1βH−1Gr,1+1φy2mr,y2mr1/p+1G1r≤φy2mr,y2mr1/p+1αφβ,β1/p+1+c1φ1+1βH−1Gr,1+1φβ,β1/p+1G1r,(41)y2mr≤β+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1hsψy1ms,y2mskds1/kdt≤β+ψy1mr,y2mr+11/p+1G2r≤β+ψy1mr,y1mr+H−1Gr+11/p+1G2r=β+ψy1mr,y1mr1+1y1mrH−1Gr+11/p+1G2r≤β+ψy1mr,y1mr1+1αH−1Gr+11/p+1G2r≤β+c2ψy1mr,y1mrψ1,1+1αH−1Gr+11/p+1G2r=ψy1mr,y1mr1/p+1βψy1mr,y1mr1/p+1+c2ψ1,1+1αH−1Gr+1ψy1mr,y1mr1/p+1G2r≤ψy1mr,y1mr1/p+1βψα,α1/p+1+c2ψ1,1+1αH−1Gr+1ψα,α1/p+1G2r.
Similar to the above, by Lemma3, N2, (40), (41), we obtain(42)rN−kky1mr′k′=rN−1CN−1k−1ℜ−1brφy1m−1r,y2m−1rk≤rN−1CN−1k−1ℜ−1brφy1mr,y2mrk≤rN−1CN−1k−1ℜ−1brφy1mr,ψy1mr,y1mr1/p+1βψα,α1/p+1+c2ψ1,1+1αH−1Gr+1ψα,α1/p+1G2rk≤rN−1CN−1k−1ℜ−1brc1φy1mr,ψy1mr,y1mr1/p+1ω1rk,(43)rN−kky2mr′k′=rN−1CN−1k−1ℜ−1hrψy1m−1r,y2m−1rk≤rN−1CN−1k−1ℜ−1hrψy1mr,y2mrk≤rN−1CN−1k−1ℜ−1hrψφy2mr,y2mr1/p+1αφβ,β1/p+1+c1φ1+1βH−1Gr,1+1φβ,β1/p+1G1r,y2mrk≤rN−1CN−1k−1ℜ−1hrc2ψφy2mr,y2mr1/p+1,y2mrω2rk.
Next, we have(44)rN−kky1mr′k′≤rN−kky1mr′k′≤rN−1CN−1k−1ℜ−1brc1φy1mr,ψy1mr,y1mr1/p+1ω1rk,(45)rN−kky2mr′k′≤rN−kky2mr′k′≤rN−1CN−1k−1ℜ−1hrc2ψφy2mr,y2mr1/p+1,y2mrω2rk.
Multiplying the last inequality in (44) by y1mr′ and the last inequality in (45) by y2mr′, we arrive at(46)y1mr′k′y1mr′≤krk−1CN−1k−1ℜ−1brc1φy1mr,ψy1mr,y1mr1/p+1ω1rky1mr′,(47)y2mr′k′y2mr′≤krk−1CN−1k−1ℜ−1hrc2ψφy2mr,y2mr1/p+1,y2mrω2rky2mr′.
ByN1 and Lemma 1, integrating (46) and (47) from 0 to r, we get(48)y1mr′k+1≤k+1k∫0rksk−1CN−1k−1ℜ−1bsc1φy1ms,ψy1ms,y1ms1/p+1ω1sky1ms′ds≤k+1CN−1k−1∫0rsk−1ℜ−1bsc1ω1skφy1ms,ψy1ms,y1ms1/p+1+1k/p+1y1ms′ds≤k+1CN−1k−1rk−1ℜ−1c1ϕ1rω1rk∫0rφy1ms,ψy1ms,y1ms1/p+1+1k/p+1y1ms′ds=k+1CN−1k−1rk−1ℜ−1c1ϕ1rω1rk∫αy1mrφs,ψs,s1/p+1+1k/p+1ds≤c1ω1r+1k/p+1k+1CN−1k−1rk−1ℜ−1ϕ1rk∫αy1mrφs,ψs,s1/p+1+1k/p+1ds,(49)y2mr′k+1≤k+1k∫0rksk−1CN−1k−1ℜ−1hsc2ψφy2ms,y2ms1/p+1,y2msω2sky2ms′ds≤k+1CN−1k−1∫0rsk−1ℜ−1hsc2ω2skψφy2ms,y2ms1/p+1,y2ms+1k/p+1y2ms′ds≤k+1CN−1k−1rk−1ℜ−1c2ϕ2rω2rk∫0rψφy2ms,y2ms1/p+1,y2ms+1k/p+1y2ms′ds=k+1CN−1k−1rk−1ℜ−1c2ϕ2rω2rk∫βy2mrψφs,s1/p+1,s+1k/p+1ds≤c2ω2r+1k/p+1k+1CN−1k−1rk−1ℜ−1ϕ2rk∫βy2mrψφs,s1/p+1,s+1k/p+1ds.
By (48) and (49), we get(50)y1mr′≤c1ω1r+1k/p+1k+1k+1CN−1k−1rk−1ℜ−1ϕ1rkk+1×∫αy1mrφs,ψs,s1/p+1+1k/p+1ds1/k+1,
(51)y2mr′≤c2ω2r+1k/p+1k+1k+1CN−1k−1rk−1ℜ−1ϕ2rkk+1×∫βy2mrψφs,s1/p+1,s+1k/p+1ds1/k+1.
From the above two inequalities, we easily deduce that(52)y1mr′∫αy1mrφs,ψs,s1/p+1+1k/p+1ds1/k+1≤c1ω1r+1k/p+1k+1k+1CN−1k−1rk−1ℜ−1ϕ1rkk+1,
(53)y2mr′∫βy2mrψφs,s1/p+1,s+1k/p+1ds1/k+1≤c2ω2r+1k/p+1k+1k+1CN−1k−1rk−1ℜ−1ϕ2rkk+1.
Integrating (52) and (53) from 0 tor, we arrive at(54)∫αy1mr1∫0μφt,ψt,t1/p+1+1k/p+1dtk+1dμ≤∫0rc1ω1s+1k/p+1k+1k+1CN−1k−1sk−1ℜ−1ϕ1skk+1ds,
(55)∫βy2mr1∫0μψφt,t1/p+1,t+1k/p+1dtk+1dμ≤∫0rc2ω2s+1k/p+1k+1k+1CN−1k−1sk−1ℜ−1ϕ2skk+1ds.
Now, the above two inequalities can be written as(56)F1y1mr≤P2r,∀r≥0,
and(57)F2y2mr≤Q2r,∀r≥0.
Finally, by the fact thatF1−1 and F2−1 are strictly increasing on 0,F1∞ and 0,F2∞ separately, we get(58)y1mr≤F1−1P2r,∀r≥0,
(59)y2mr≤F2−1Q2r,∀r≥0.
Then, we prove that the sequencesy1mrm≥0 and y2mrm≥0 are bounded on 0,c0 for arbitrary c0>0. Indeed, since(60)y1mr′≥0,y2mr′≥0,∀r≥0.
It follows that(61)y1mr≤y1mc0≤C1,y2mr≤y2mc0≤C2,on0,c0,where C1=F1−1P2c0 and C2=F2−1Q2c0 are positive constants. Moreover, from (50) and (51), we can deduce that y1mr′m≥0 and y2mr′m≥0 are bounded on 0,c0 for arbitrary c0>0. Thus, the sequences y1mrm≥0 and y2mrm≥0 are bounded and equicontinuous on 0,c0 for arbitrary c0>0. By Arzela–Ascoli theorem, there exist subsequences of y1mrm≥0 and y2mrm≥0 converging uniformly to y1r and y2r on 0,c0. Since y1mrm≥0 and y2mrm≥0 are increasing on 0,∞, then y1mrm≥0 and y2mrm≥0 converge uniformly to y1r and y2r on 0,∞. By the arbitrariness of c0, we deduce that y1,y2 is an entire positive radial solution of system (32). Thus, by Lemma 3, we get that z1,z2 is an entire positive radial solution of system (8). Repeating the proof in [1], we get y1,y2∈C20,∞×C20,∞. Thus, by Lemma 3, we deduce that system (8) has an entire positive radial solution z1,z2∈C20,∞×C20,∞. This completes the proof.Theorem 2.
We assumeN1, N2, S1, and S2 hold, then system (8) has an entire positive bounded radial solution z1,z2 such that(62)α+P1r≤z1r≤F1−1P2r,β+Q1r≤z2r≤F2−1Q2r.Proof.
On the basis ofN1, N2, by a proof similar to one of Theorem 1, it is easy to prove that (8) has an entire positive radial solution z1,z2∈C20,∞×C20,∞. Moreover, it follows from (56), (57), and S1 that(63)F1y1mr≤P2∞<F1∞<∞,∀r≥0.
(64)F2y2mr≤Q2∞<F2∞<∞,∀r≥0.
SinceF1−1 and F2−1 are strictly increasing on 0,F1∞ and 0,F2∞ separately, we get(65)y1mr≤F1−1P2∞<∞,∀r≥0,
(66)y2mrF2−1Q2∞<∞,∀r≥0.
Lettingm⟶∞ in the above two inequalities, we obtain(67)y1r≤F1−1P2∞<∞,∀r≥0,
(68)y2r≤F2−1Q2∞<∞,∀r≥0.
Lettingm⟶∞ in (35), we get(69)y1r=α+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bsφy1s,y2skds1/kdt,
(70)y2r=β+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1hsψy1s,y2skds1/kdt.
Then, it follows fromN1, N2, S2, and Lemma 1 that(71)y1r=α+∫0rktN−k∫0tsN−1CN−1k−1ℜ−1bsφy1s,y2skds1/kdt≥α+∫0rkνN−k∫0vtN−1CN−1k−1ℜ−1btφα,β+∫otkμN−k∫0μsN−1CN−1k−1ℜ−1hsψy1s,y2skds1/kdμkdt1/kdv≥α+∫0rkνN−k∫0νtN−1CN−1k−1ℜ−1btφα,β+1ψα,β+11/p+1G2tkdt1/kdν=α+P1r.
As in the preceding lines, we can prove(72)y2r≥β+Q1r.
This completes the proof.


## 4. Entire Positive Blow up Radial Solution

In this section, we investigate entire positive blow up radial solution of system (8), and the main result is as follows:Theorem 3.
We assume thatN1, N2, S3, and S4 hold, then the nonlinear k-Hessian system (8) has an entire positive blow up radial solution z1,z2∈C20,∞×C20,∞.Proof.
It follows from the conditionsN1, N2 that a similar proof of Theorem 1 ensures that system (8) has an entire positive radial solution z1,z2∈C20,∞×C20,∞. Moreover, it follows from (56) and (57) that(73)F1y1mr≤P2∞,∀r≥0,(74)F2y2mr≤Q2∞,s∀r≥0.
SinceF1−1 and F2−1 are strictly increasing on 0,F1∞ and 0,F2∞ separately, we get(75)y1mr≤F1−1P2∞,∀r≥0,(76)y2mr≤F2−1Q2∞,∀r≥0.
WhenS3 holds, we see that F1−1∞=F2−1∞=∞. By the condition S3, letting m⟶∞ in the above two inequalities, we obtain(77)y1r≤F1−1P2∞,∀r≥0,(78)y2r≤F2−1Q2∞,∀r≥0.
By the conditionS3, letting r⟶∞ in the above two inequalities, we obtain(79)limr⟶∞y1r≤F1−1P2∞=∞,∀r≥0,(80)limr⟶∞y2r≤F2−1Q2∞=∞,∀r≥0.
Then, it follows fromS4, (72), and (73) that(81)limr⟶∞y1r≥α+limr⟶∞P1r>P1∞=∞,(82)limr⟶∞y2r≥β+limr⟶∞Q1r>Q1∞=∞.
Consequently,(83)limr⟶∞y1r=∞,limr⟶∞y2r=∞.
Therefore, system (33) has an entire positive blow up radial solution y1,y2∈C20,∞×C20,∞. By Lemma 3, system (8) has an entire positive blow up radial solution z1,z2∈C20,∞×C20,∞. The proof is completed.


## 5. Semifinite Entire Positive Blow up Radial Solution

In this section, we investigate semifinite entire positive blow up radial solution of system (8), and the main results are as follows:Theorem 4.
We assume thatN1, N2, S5, and S6 hold, then the nonlinear k-Hessian system (8) has one semifinite entire positive blow up radial solution z1,z2∈C20,∞×C20,∞.Proof.
In view ofN1, N2, the same arguments as in Theorem 1, we can know that system (8) has an entire positive radial solution z1,z2∈C20,∞×C20,∞. By S5, (79), and (81), we get(84)limr⟶∞y1r≤F1−1P2∞=∞,(85)limr⟶∞y1r≥α+limr⟶∞P1r>P1∞=∞,
which imply that(86)limr⟶∞y1r=∞.
Moreover, byS6, (69), and (73), we obtain(87)limr⟶∞y2r≤F2−1Q2∞<∞,(88)limr⟶∞y2r≥β+limr⟶∞Q1r>Q1∞,Q1∞<∞,
which imply that(89)limr⟶∞y2r<∞.
Therefore, system (33) has a positive semifinite entire blow up radial solution y1,y2∈C20,∞×C20,∞. By Lemma 3, system (8) has a positive semifinite entire blow up radial solution z1,z2∈C20,∞×C20,∞. The proof is finished.Theorem 5.
We assume thatN1, N2, S7, and S8 hold, then the nonlinear k-Hessian system (8) has a positive semifinite entire blow up radial solution z1,z2∈C20,∞×C20,∞.Proof.
Same as Theorem4, we can know that system (8) has an entire positive radial solution z1,z2∈C20,∞×C20,∞. By S7, (81), and (83), we get(90)limr⟶∞y2r≤F2−1Q2∞=∞,(91)limr⟶∞y2r≥β+limr⟶∞Q1r>Q1∞=∞,
which imply that(92)limr⟶∞y2r=∞.
Moreover, byS8, (68), and (72), we obtain(93)limr⟶∞y1r≤F1−1P2∞<∞,(94)limr⟶∞y1r≥α+limr⟶∞P1r>P1∞,P1∞<∞,
which imply that(95)limr⟶∞y1r<∞.
Therefore, system (33) has a positive semifinite entire blow up radial solution y1,y2∈C20,∞×C20,∞. By Lemma 3, system (8) has a positive semifinite entire blow up radial solution z1,z2∈C20,∞×C20,∞. The proof is finished.


## 6. Example

Example 1.
we consider(96)ΛS41/4λD2z1S41/4λD2z1=142−xx4exz11/3z22/3,x∈ℝ6,ΛS41/4λD2z2S41/4λD2z2=342−2xx4e2xz11/2z21/2,x∈ℝ6.
LettingΛs=s3, p=3, then Λ∈Θ. Here, bs=2−s/s4es, hs=2−2s/s4e2s, φz1,z2=1/4z11/3z22/3, φz1,z2=3/4z11/3z22/3, then φ and ψ are increasing for each variable which satisfies N1. Obviously, when α=β=4, we have t1≥4, s1≥1, t2≥34, s2≥1, t3≥1, s3≥1,(97)φt1s1,t2s2=14t11/3s11/3t22/3s22/3≤c114t11/3t22/314s11/3s22/3=c1φt1,t2φs1,s2,∀c1≥4,(98)ψt3s3,t1s1=34t31/2s31/2t11/2s11/2≤c234t31/2t11/234s31/2s11/2=c2ψt3,t1ψs3,s1,∀c2≥43,(99)φα,β≥5−12,ψα,β≥5−12.which mean that N2 is satisfied. By Theorem 1, the 4-Hessian system (97) has an entire positive radial solution z1,z2∈C20,∞×C20,∞.Example 2.
we consider(100)ΛS41/4λD2z1S41/4λD2z1=x511z1z24,x∈ℝ6,ΛS41/4λD2z2S41/4λD2z2=x410z13+z22,x∈ℝ6.
LettingΛs=s3, p=3, then Λ∈Θ. Here bs=11s5, hs=10s4, φz1,z2=z1z24, ψz1,z2=z13+z22, then φ and ψ are increasing for each variable which satisfies N1. Obviously, when α=β=1, we have t1≥1, s1≥1, t2≥1, s2≥1/24, t3≥1, s3≥1,(101)φt1s1,t2s2=t1s1t24s24≤c1t1t24s1s24=c1φt1,t2φs1,s2,∀c1≥1,(102)ψt3s3,t1s1=t33s33+t12s12≤c2t33+t12s33+s12=c2ψt3,t1ψs3,s1,∀c2≥1, (103)φα,β≥5−12,ψα,β≥5−12.
which mean thatN2 is satisfied. After a simple calculation, one has(104)P2∞=∫0∞c1ω1s+11/5125s3ϕ1sds>∫0∞12s85ds=125∫0∞s8/5ds=∞,(105)Q2∞=∫0∞c2ω2s+11/5125s3ϕ2sds>∫0∞12s75ds=125∫0∞s7/5ds=∞,(106)F1∞=∫α∞1∫0μt4+t3+1dt5dμ=∫α∞11/55μ5+1/4μ4+μdμ>∫α∞1μ+155dμ=∫α∞1μ+1dμ=∞,(107)F2∞=∫β∞1∫0μt15/4+t2+1dt5dμ=∫β∞14/19μ19/4+1/3μ3+μ5dμ>∫β∞1μ+155dμ=∫β∞1μ+1dμ=∞,
which mean thatS3 is satisfied. Then, we have(108)G1r=∫0r4t2∫0ts5C5311s5ds1/4dt=254∫0rt9/4dt=413254r13/4,(109)G2r=∫0r4t2∫0ts5C5310s4ds1/4dt=254∫0rt2dt=13254r3,(110)P1∞=∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1btφα,β+1ψα,β+11/p+1G2tkdt1/kdν>∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1btφα,βkdt1/kdν>∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1bt1φα,β+1kdt1/kdν>1φα,β+11/p+1∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1btkdt1/kdν=11+11/4G1∞=∞,(111)Q1∞=∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1htψα+1φα,β+11/p+1G1t,βkdt1/kdν>∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1htψα,βkdt1/kdν>∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1ht1ψα,β+1kdt1/kdν>1ψα,β+11/p+1∫0∞kνN−k∫0νtN−1CN−1k−1ℜ−1htkdt1/kdν=12+11/4G2∞=∞,
which mean thatS4 is satisfied. By Theorem 3, the 4-Hessian system (101) has an entire positive blow up radial solution z1,z2∈C20,∞×C20,∞.


## 7. Conclusion

Thek-Hessian equations, a deep extension of the Laplace equation, have played an important role in mathematics and other applied sciences. In this paper, we studied the radial solutions of a class of k-Hessian system involving a nonlinear operator. The Keller–Osserman type conditions are also crucial throughout the proof process. By employing the monotone iterative method, we establish some new existence results on positive radial solutions of the k-Hessian system (8) in cases of finite, infinite, and semifinite. Our results complement the work in [1] and extend the existing results [7, 8, 15–17, 38], which is a meaningful contribution to the topic of nonlinear elliptic system.



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*Source: 6681813-2022-08-10.xml*