(高承華) (何興玥) (王晶晶)

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

E-mail: gaokuguo@163.com; hett199527@163.com; WJJ950712@163.com

Abstract In this paper,we focus on the following coupled system of k-Hessian equations:Here B is a unit ball with center 0 and fi(i=1,2) are continuous and nonnegative functions.By introducing some new growth conditions on the nonlinearities f1 and f2,which are more flexible than the existing conditions for the k-Hessian systems (equations),several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.

Key words system of k-Hessian equations; k-convex solutions;existence;multiplicity;fixed-point theorem

1 Introduction

In this paper,we aim to investigate the existence and multiplicity for the coupled system ofk-Hessian equations

wherek=1,2,···,N,B={x ∈RN:|x|<1} is a unit ball with center 0,fi ∈C([0,1]×[0,+∞),[0,+∞)),and are not identical zeros,i=1,2.

In general,thek-Hessian operatorSkis defined as

which is the sum of allk×kprincipal minors of the Hessian matrix ofD2u,whereλ(D2u)=(λ1,λ2,···,λN) is the vector of eigenvalues ofD2u,andλ1,λ2,···,λNare the eigenvalues of the Hessian matrix[5,28].From the terms of divergence,Sk(λ(D2u))=;for more details see [14,24].It is noteworthy that thek-Hessian operators are fully non-linear whenk ≥2.These include the classical Laplace operator ?uwhenk=1 and the Monge-Ampère operator det(D2u) whenk=N.It is wellknown that the existence of positive radial solutions to these kinds of two types of problem have been discussed by several authors and has many excellent results have been obtained,see,for instance,[1,16,18,32]and the references therein.

It is well known thatk-Hessian equation is the classical fully nonlinear partial differential equation,and it has lots of applications in geometry,fluid mechanics and other applied disciplines.The study of thek-Hessian equation has also attracted the attention of many scholars;see [2–4,6,7,11–14,18–26,29,30,32,33],and the references therein.Various results for solution of thek-Hessian equation have been obtained by using different approaches and techniques,for instance,C1,1solutions [25],C2+αlocal solutions [2],blow-up solutions [22,31,33,34]and other results.Meanwhile,the existence of radial solutions concerning a single equation has been widely investigated.For recent advances on this subject,see Covei [4],Feng and Zhang[11,12],Sánchez and Vergara [21],Wei [26,27],Zhang [30]and the references therein.

It is noted that if we takek=N,f(|x|,-v)=(-v)αandf(|x|,-u)=(-u)β,then system(1.1) is reduced to a special power-type Monge-Ampère system

By using the fixed-point theory in cones and the eigenvalue theory for the Monge-Ampère operator,Zhang and Qi [35]discussed the existence,uniqueness and nonexistence of the radial convex solutions to problem (1.2).Later,this result was generalized by Liuet al.[18]for a Monge-Ampère system with more general nonlinearity asf1(|x|,-v) andf2(|x|,-u) (in fact,Liuet al.[18]also considered a more general system of Monge-Ampère).Feng [10]considered a Monge-Ampère coupled system withnequations andnpositive parameters,and obtained some new existence results by decoupling composite operators and using the eigenvalue theory in cones.Moreover,he also analyzed the asymptotic behavior of nontrivial radial convex solutions to the system.Meanwhile,Gaoet al.[13]extended the results of Qi and Zhang [35]to ak-Hessian system with the same nonlinearities,and they obtained similar results.Furthermore,Feng and Zhang[11]used the eigenvalue theory in cones to obtain the existence,multiplicity,and parameter dependence of nontrivial radial solutions to a kind of autonomick-Hessian system with parameters.Heet al.[15]used the fixed-point theorem in cones to obtain the existence and nonexistence of radialk-convex solutions for a generalk-Hessian system.

Motivated by the above results,using the well-known fixed-point theorem in cones,we try to obtain the existence and multiplicity of radialk-convex solutions to system (1.1).

Then,under some different suitable conditions imposed on(here,we may call these theαiorβi-asymptotic growth condition,the super-αiorβi-asymptotic growth condition,or the sub-αiorβi-asymptotic growth condition),as well as some properties of inequalities imposed onαiandβi,we obtain the existence of radialk-convex solutions to system (1.1);see,for instance,Theorems 3.1–3.4.It is noted that the asymptotic growth conditions on the nonlinearitiesfiin the existing results,like [11,19]and [30],are ofk-asymptotic growth or super-or sub-k-asymptotic growth,where if is the case that the constants areαi=kandβi=k.Therefore,our conditions here are more flexible than those of the existing results,and the results here are completely new.

The rest of this paper is organized as follows: in Section 2,we construct a composite operator for thek-Hessian system and discuss the properties of this operator in a given positive cone.In Section 3,we show the existence of radialk-convex solutions for a coupled system(1.1),when the nonlinear terms satisfy different and new growth conditions,and prove these by overcoming the difficulties caused by the composite operator.In Section 4,we show and prove some multiplicity results for radialk-convex solutions for a coupled system (1.1).In Section 5,we present some numerical examples to illustrate our main results.

2 Preliminary Results on Radial Solutions

Based on this,by using a shift transformation asu=-?1andv=-?2,system (1.1) can be transformed into the following boundary value problem for the sake of simplicity,we still useuandvhere

Then,by integration,we can obtain that

LetXbe the Banach spaceC[0,1]equipped with the supermum normand letK ?Xbe a cone defined as follows:

Let ?R={v ∈K;‖v‖

Sincefi: [0,1]×[0,+∞)→[0,∞) are continuous and are not identical zero,we can see that operatorsTi:K →K(i=1,2) are completely continuous operators.Thus,T:K →Kis also a completely continuous operator.Thenuis a fixed-point ofTif and only ifuis a positive solution to problem (2.1).Furthermore,uis a radialk-convex solution of system (1.1).

Let

Now we give some Lemmas.

Lemma 2.1Letηi>0.If,for anyu,v ∈Kandτ ∈[0,1],f1(τ,v(τ))≥η1va(τ) andf2(τ,u(τ))≥η2ub(τ),then

ProofSinceu,v ∈K,we obtain that

Therefore,(2.3) holds.Similarly,we can obtain that (2.4) holds.

Lemma 2.2Letεi>0.If,for anyu,v ∈Kandτ ∈[0,1],f1(τ,v(τ))≤ε1vc(τ) andf2(τ,u(τ))≤ε2ud(τ),then

ProofSinceu,v ∈K,we can obtain that

Thus,(2.5) is correct.Similarly,(2.6) holds.

Our main tools depend on analysis methods and on the well-known results from the fixedpoint theorem.

Lemma 2.3([8,17]) LetEbe a Banach space and letK ?Ebe a cone.Assume that?1,?2are bounded,open subsets ofEwithθ ∈?1,??2,and letA:K ∩(?1)→Kbe a completely continuous operator such that either

(i)‖Au‖≤‖u‖,u ∈K ∩??1,and‖Au‖≥‖u‖,u ∈K ∩??2;or

(ii)‖Au‖≥‖u‖,u ∈K ∩??1,and‖Au‖≤‖u‖,u ∈K ∩??2.

ThenAhas a fixed-point inK ∩(?1).

3 Existence Results for k-convex Solutions

Now,we establish the existence results of thek-convex solutions for the coupled system(1.1) for the nonlinearityfisatisfies different growth conditions.

Theorem 3.1Suppose that∈(0,+∞),∈(0,+∞) andf2(τ,0)=0.If the constants areαi,βi>0 (i=1,2) with

then system (1.1) has at least one radialk-convex solution.

ProofIn view of the definitions of(i=1,2),there always exists a positive constantr1∈(0,1) such that,forτ ∈[0,1],

whereε1is chosen such that 0<ε1<(i=1,2).

Then,by the assumption thatf2(τ,0)=0 and the continuity off,there exists another constantr2:0

Now,foru ∈K ∩??r2,it follows from Lemma 2.2 and (3.3) that

Therefore,for anyu,v ∈[0,r1],combining Lemma 2.1 with (3.1) and (3.2),we get that

Moreover,by the definition of the operatorT,for anyu ∈K ∩??r2,we get that

Sinceα1α2

This implies that

On the other hand,it follows from∈(0,+∞) that,for eachτ ∈[0,1],there exist positive constantsR1andε2such that

Then we can obtain that

Furthermore,it follows from Lemma 2.2 and (3.6) that

and combining this with (3.6),we have

Let

Combining the above inequalities withβ1β2

This shows that

Hence,from Lemma 2.3,we know thatThas at least one fixed-point inK ∩(?r2).

Similarly to the proof of Theorem 3.1,with some necessary modifications,we have following results:

Theorem 3.2Assume that we have the constantsαi,βi>0(i=1,2) with

Then system (1.1) has at least one radialk-convex solution iff2(τ,0)=0 and if one of the following conditions is satisfied:

Theorem 3.3Suppose thatIf we have the constantsαi,βi>0 (i=1,2) with

then (1.1) has at least one radialk-convex solution.

ProofFor anyη1>0,let

Then,for anyu ∈K ∩??r4,it follows from Lemma 2.2 that

Now,for anyu,v ∈[0,r3],it follows from Lemma 2.2 that

Hence,for anyu ∈K ∩??r4,we can deduce that

Sinceα1α2>k2,similarly to the previous method,we can get that

This implies that

In addition,it can be obtained from∈(0,∞) that there exists a constantR3>1 such that,for anyτ ∈[0,1],

Combining this with (2.4),foru ∈K ∩??R4,we have

Now,sincev=T2u ∈K,foru ∈K ∩??R4,we also get that

Therefore,we can deduce that

Sinceβ1β2>k2,it follows from

that

From Lemma 2.3,we know thatThas at least one fixed-point inK ∩(?r3).

Similarly to the proof of Theorem 3.3,with some necessary modifications,we have following result:

Theorem 3.4Assume that we have the constantsαi,βi>0 (i=1,2) satisfying that

Then system (1.1) has at least one radialk-convex solution if one of the following conditions is satisfied:

4 Multiplicity Results for Radial Solutions

Theorem 4.1Suppose that∈(0,+∞),∈(0,+∞) and thatf2(τ,0)=0.If we have constantsαi,βi>0 (i=1,2) with

and there exists a constant ^rsuch that

then (1.1) has at least two radialk-convex solutions.

ProofBy the definition of,for anyu ∈K ∩,it follows from Lemma 2.2 that

This implies that

Sinceα1α2k2,combining Theorem 3.1 with Theorem 3.3,we have that there exist a sufficient small constantr2>0 and a sufficient large constantR4>0 such that

Combining this and Lemma 2.3,we get thatThas at least two-fixed points inK ∩(?r2)andK ∩().

Theorem 4.2Suppose that∈(0,+∞),∈(0,+∞) and thatf2(τ,0)=0.If we have constantsαi,βi>0(i=1,2) with

then (1.1) has at least two radialk-convex solutions.

ProofBy the definition ofand Lemma 2.2,for anyu ∈K ∩,we have that

This implies that

Sinceα1α2>k2,β1β20 and a sufficient smallr4>0 such that

Combining this with Lemma 2.3,we know thatThas at least two-fixed points inK ∩() andK ∩(?r4).

By making some necessary modifications to the proofs of Theorems 4.1 and 4.2,we can obtain the following conclusions:

Theorem 4.3Assume that we have the constantsαi,βi>0 (i=1,2),with

Then system (1.1) has at least two radialk-convex solutions iff2(τ,0)=0 and if one of the following conditions is satisfied:

Theorem 4.4Assume that we have the constantsαi,βi>0 (i=1,2),with

Then system (1.1) has at least two radialk-convex solutions iff2(τ,0)=0 and if one of the following conditions is satisfied:

5 Numerical Examples

Now we show some numerical examples to illustrate our main results.For the sake of simplicity,we only give the examples to illustrate the multiplicity results.

Consider the boundary value problem

whereN=3,k=2.

Example 5.1In problem (5.1),we take the constantsandα1=1,α2=2,β1=5,β2=3 withα1α2k2,and choose the functions

By calculation,we obtain that

Furthermore,

which implies that if the conditions of Theorem 4.1 hold,then problem (5.1) has at least two radialk-convex solutions.

Example 5.2In problem (5.1),we take the constantsandα1=3,α2=6,β1=2,β2=withα1α2>k2,β1β2

By calculation,we obtain that

which implies that if the conditions of Theorem 4.2 hold,then problem (5.1) has at least two radialk-convex solutions.

Conflict of InterestThe authors declare no conflict of interest.