k -convex solutions for multiparameter Dirichlet systems with k -Hessian operator and Lane-Emden type nonlinearities

: In this article, our main aim is to investigate the existence of radial k -convex solutions for the following Dirichlet system with k -Hessian operators:

Here, u v p q 1 1 is called a Lane-Emden type nonlinearity.The weight functions )) with > < ν r ν r 0 1 2 ( ) ( ) for all ∈ r R 0, ( ], p q , 1 2 are nonnegative and q p , 1 2 are positive exponents, = R ( ) } , in general, we denote by Ω a finite ball of N around the origin.For ∈ u C Ω 2 ( ), the k-Hessian operator S D u k 2

(
) is defined as follows: )) ) is the vector of eigenvalues of the Hessian matrix D u 2 [7,46] and σ k is the k-th elementary symmetric polynomial in N variables.It is not difficult to find that when ≥ k 2, k-Hessian operator is a completely nonlinear partial differential operator, and it is a set of operators, including the Laplace operator u Δ when = k 1, and the Monge-Ampère operators Δ u det 2  ( ) when = k N , and other well-known operators.This family of operators has been studied extensively, see, for instance, [14,27,32,[52][53][54] and the references therein.The theory of completely nonlinear equations involving k-Hessian operators was originally developed by Caffarelli, Nirenberg, Spruck [7,38,[42][43][44], Jacobsen [29], Wang [46], and others in the classical context.We refer to [4,7,19,25,28,38,46] for these and further results.In recent years, many articles were devoted to the study of the Dirichlet problem for a single equation with k-Hessian operator For this problem, Wang [47] studied the existence of classical solutions determined by the Hessian eigenvalue function and dealt with the corresponding functional, and obtained an inequality similar to that of the Sobolev inequality.And what is even more interesting is for different forms of the nonlinear term f λ u , ( ), in [29], the author discussed different existence results and obtained the global bifurcation phenomena for this problem.A broader discussion of this problem can also be found in [6,7,[39][40][41]45] and so on.Over the past decade, a number of scholars have developed a keen interest in the aforementioned problems with singular or nonsingular linear terms, like u p , u x u x ln k p ).They used the fixed point theory, variational method; in this article, the existence and asymptotic behavior of different solutions for this kind of problem are discussed, such as blow-up solution, convex solution, and bounded solution, see for instance [12,13,16,33,37,50,51].
At the same time, the Lane-Emden type nonlinear term attracts our attention, since it is widely used to investigate the theory of stellar structure, the thermal behavior of spherical cloud gas, and the theory of thermionic currents [8,9,36].It is worth noting that Lane-Emden type plays an important role in various kinds of nonlinearities, and its specific form is + k u kv p q 1 2 [11,48] or k u v α β 3 [21,31].Here, we are more focused on the second case.There are many excellent results on the existence of solutions of equations with nonlinear term.In the case of a single equation, such nonlinearity has already been considered in the Dirichlet problem with the Laplace operator and mean curvature operator, respectively, see [1,3,5,10,15,34,49] and references therein.In 2020, Farina and Hasegawa [15] obtained the non-existence and existence of the stable solution for the weighted Lane-Emden type equations by proving a Liouville-type theorem, where

(
), and . Moreover, he also assumed the weight ) with > W 0 almost everywhere in m .In particular, when the weight function = W 1, Damascelli et al. [10] considered the sign changing solutions of the equation with m-Laplace operator −Δ u x m ( ) in possibly unbounded domains or in N .Here > m 1 and u is a possibly unbounded function which may change sign.Different from [15], they gave the critical exponent for the existence of the solution of parameter p.More specifically, Zhang and Zhu [49] investigated the existence and multiplicity of solutions for a class of sub-elliptic systems with Hardy-type potential and multiple critical exponentials on Carnot groups by means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz.This is a very interesting result.
The study of systems with Lane-Emden type nonlinearity seems to be at the beginning.We note the recent articles [2,3].Corsato et al. [3] examined the system given by where λ γ , are positive parameters, ≤ ≤ p θ 1 , and Ω is a smooth bounded domain in N .They showed that the extremal solutions associated with the above system are bounded.In addition, nonlinear terms of Lane-Emden type are gradually studied in other operator system problems, such as in 2019, Gurban and Jebelean [22] dealt with Dirichlet systems involving the mean curvature operator and Lane-Emden type nonlinearities in Minkowski space is called the mean curvature operator, p 1 , q 2 are nonnegative, while q 1 , p 2 are positive exponents, weight functions ) are assumed to be continuous with By using the upper and lower solution method and the Leray-Schauder degree theory, they obtained the existence, non-existence, and multiplicity of radial solutions dependent on parameters λ 1 and λ 2 .It is worth mentioning that when u v p q ), Gurban and Jebelean et al. discussed this problem in [23,24] and the above results also hold when f x u v , , (| | ) meets appropriate conditions.However, to our best knowledge, there are few research studies on k-Hessian operators with nonlinear terms of Lane-Emden type nonlinearity.It appears as a natural direction the study of systems involving the k-Hessian operator and Lane-Emden type nonlinearities where weight functions ], p q , 1 2 are nonnegative and q p , 1 2 are positive exponents.Before we consider problem (1.1), we first consider the Dirichlet problem with a general nonlinear term Here the nonlinear functions f 1 and )are continuous.We note that the study of the existence of solutions for (1.2) type problems have many interesting results.Based on the fixed-point index theory in cones, Gao et al. [20] considered a special class of nonlinear terms, which are given as , are positive constants), and the authors obtained the existence, uniqueness, and nonexistence of radial convex solutions for some suitable constants α and β.Further, Feng and Zhang [17] studied a kind of auto- nomic system with positive parameters and nonlinear terms of ( ) type and obtained the existence, multiplicity, and parameter dependence of nontrivial radial solutions by using the eigenvalue theory in cones.However, it is worth noting that in reference [17], the authors put a strict limit on the range of k, namely ∕ < < N k N 2 .In addition, Zhang and Zhou [55] investigated problem (1.2) with positive weights and nonlinear terms of p x f v (| |) ( ), q x g u (| |) ( ) type, f g , are continuous and increasing functions.The existence of entire positive k-convex radial solutions is gained in this condition.Furthermore, Covei [4] gave a necessary and sufficient condition for the existence of positive radial solutions when f g , meet some other hypotheses.Inspired by the aforementioned works, in Section 2, we first obtain the existence of nontrivial solutions of problem (1.2) and consider the monotonicity of solutions according to the monotonicity of nonlinearities.Then, we prove that problem (1.2) still admits k-convex solution and the solution is strictly increasing when problem (1.2) with positive two-parameter.In Section 3, by using the Leray-Schauder degree theory, we discuss the existence and monotonicity of k-convex solutions for problem (1.2) when the nonlinear term f i is under superlinear and sublinear conditions, respectively.In Section 4, by applying upper and lower solution methods, we construct a suitable upper and lower solution and auxiliary functions of problem (1.2), and gain the existence of the k-convex solution by combining Leray-Schauder degree theory.
Based on these preparations, in Section 5, at first, we make the following hypothesis: , and < <∞ N p q , 1 2 . We give the main results for problem (1.1): and a continuous function Here, the set 1 is adjacent to the coordinate axes λ 0 1 and λ 0 2 , and the curve is asymptotically close to two straight lines parallel to the coordinate axes λ 0 1 , λ 0 2 .The difficulties arising in studies of such problems with Lane-Emden type nonlinear terms and a unbounded operator are well known.We need to find a bounded domain in which the k-Hessian operator can reach its maximum or minimum, which requires us to construct a cone very well when we use the Leray-Schauder degree theory.In the process of constructing the upper and lower solution, we should also consider the boundedness of the upper and lower solutions.
Throughout this article, we denote that will be endowed with We denote ( ) be a circle with radius ρ and with its center at the origin, ρ is a constant greater than R.Then, ρ ( ) is a bounded closed domain and for any ∈ u v ρ , ( ), the nonlinear term = f i 1, 2 i ( ) can reach its maximum, that is, we can write

Preliminaries
We first consider problem (1.2).To seek radial solutions of (1.2), as usual, for radial solution = u x u x ( ) (| |) and with a shift transformation as = − u ψ, = − v ϕ, system (1.2) can be reduced to the following boundary value problem, for the sake of simplicity, but we still use u and v here: , 0 0 0 .
We define a couple of nonnegative functions In addition, we say that ) of (2.1), we understand both u and v are positive.Two linear operators → C C : and → C C : 1 are given as It is not hard to see that is compact and is bounded.Therefore, we see that the nonlinear operator ) is as follows: )are continuous and map a bounded set to a bounded set.A couple of functions u v , ( ) is a solution of (2.1) if and only if it is a fixed point of the compact nonlinear operator In the following statement, we denote the Leray-Schauder degree by d LS .
Based on this, we have the following results: Proof.From the above definitions of operators, we know that which shows that (2.2) holds from the aforementioned inequalities.
Then, the following formula can be obtained from the homotopy invariance of the Leray-Schauder degree: Next, we choose some constants and introduce a continuous function → ϕ P : defined as follows: and set ) . Proof. then At first, we consider the homotopy given as follows: For this end, we claim that which is a contradiction.In addition, we know that = ϕ u v α , 0 0 ( ) also contradicts with (2.3).On the contrary, we assume that Then, by the invariance under homotopy of the Leray-Schauder degree, we infer that Then, there exists , we obtain the contradiction Proof.For any Thus, for any ) is said to be quasi-monotone nondecreasing with respect to t (resp.s) if for fixed r s , (resp.r t , ) one has ) are quasi-monotone nondecreasing with respect to both s t , , together with Then, problem (2.1) has a nontrivial solution.
Proof.We claim that (2.4) holds.Suppose on the contrary there exist ).Without loss of generality, we may assume that . Then, for fixed ∈

5). □
The following two theorems discuss the existence and monotonicity of positive solutions for problem (2.1).At first, we make the following assumptions: ) is a nontrivial solution of problem (2.1).( 1) and either u or v is positive and strictly decreasing. ( )) is quasi-monotone nondecreasing with respect to t (resp.s), ) is a positive solution with both u and v strictly decreasing. Proof.
(1) Since it means that u is decreasing.Similarly, one obtains that v is decreasing.Then = u R 0 ( ) implies that ≥ u 0 and analogously, ≥ v 0. In this regard, if ≡ u 0, we have ) ) is a positive solution.For this, we suppose that u is positive and we need to verify that v is also positive.
. And further, from the assumption that f r s t , , 2 ( )is quasi-monotone nondecreasing with respect to s, for fixed and Hence, v is strictly decreasing.Similarly, u is strictly decreasing.
) is a nontrivial solution of problem (2.1), then for any is a positive solution with both u and v strictly decreasing.
that ′ < v 0, hence, v is strictly decreasing.Similarly, we have u is strictly decreasing.
In addition, ) is a positive solution on R 0, [ ). □ Next, we consider the existence of positive solutions of problem (2.1), which depends on the parameters > < λ λ 0 , 0 0 0 .
We make the hypothesis: )are continuous, quasi-monotone nonde- creasing with respect to both s t , and satisfy ( ) is satisfied, then problem (2.7) has at least one positive solution.
Proof.In Theorem 2.6, we replace λ f i i instead of ), then it is worth noting that for any has a nontrivial radial solution.
In addition, if either ) is satisfied, then problem (2.8) has at least one radial solution u v , ( ) with both u and v are strictly increasing.
Next, we give a case of two-parameter problem involving Lane-Emden type nonlinearities to verify our results: Remark 2.11.Let p q p q , , , 1 1 2 2 be nonnegative exponents and the weight functions From the aforementioned results, we know that there exist > < λ λ * 0 * 1 2 such that for all > λ λ* ) with both u and v strictly increasing.
k-Convex solutions for k-Hessian systems  9 3 Sub-or superlinear nonlinearities near origin In this section, we focus on the existence of positive solutions to problem (2.1) when f 1 (resp.f 2 ) with sub-or superlinear growth near origin with respect to u (resp.v).
(1) ) with both u and v are strictly decreasing.
Proof.We first consider an auxiliary problem

N k k
) is a nontrivial solution of (3.3) with = μ μ i , for all ∈ i k 1, 2, …, { } .From Theorem 2.7, we know that either u i or v i is positive and strictly decreasing.Without loss of generality, we may assume that u i is positive for all ∈ i k 1, 2, …, { }(when v i is positive, the conclusion also holds).
Choose a constant > m 0 such that Then, it follows from (3.1) that we can seek a ,0 for all 0, and .
Further, integrating the first equation in (3.3) over r 0, [ ] with = u u i , = v v i , = μ μ i , using (3.5) and the fact that f r s t , , 1 ( )is quasi-monotone nondecreasing with respect to t, we have Integrating the above inequality on Then, combining the facts that u i is strictly decreasing on R 0, [ ] and > u 0 , .
Further, for all )) holds.It follows from the invariance under homotopy of the Leray-Schauder degree that
then the following boundary value problem admits a positive solution.
) are continuous and )) is quasi-monotone nondecreasing with respect to t (resp.s) and (3.1), (3.2) hold, then system (1.2) ) with either u or v is strictly increasing.In addition, if ) with both u and v are strictly increasing.
If there exists some > l 0 such that either then there exists Assume that (3.7) holds (similar reasoning when (3.8) holds), then there exists > s 0 Consider the compact homotopy We will show that there exists By contradiction, we assume that } .From Theorem 2.8, we know that both u i and v i are strictly positive on R 0, [ ).We assume that …, .
which is a contradiction.Then it follows from the invariance under homotopy of the fixed point index that )are continuous with H f 2 ( ).If (3.7) and (3.8) hold, (2.1) admits a positive solution.Further, system (1.2) admits a radial k-convex solution u v , ( ).
4 Lower and upper solutions: degree estimations In this section, we obtain some degree estimates by using the lower and upper solution method, and further obtain the existence of solutions for problem (2.1).First, we give the definitions of upper and lower solutions, respectively.
A lower solution of (2.1) is a couple of nonnegative functions An upper solution of (2.1) is a couple of nonnegative functions Let ) and an upper solution β β , )) is quasi-monotone nondecreasing with respect to t (resp.s).Then, )as follows: , 0 0 0 .
), then ′ = ′ α r u r u 0 0 ( ) ( ) and there exists a sequence ⊂ r r 0, and, it derives that Further, combining the fact α α , u v ( ) is a lower solution of (2.1) with f 1 is quasi-monotone nondecreasing with respect to t, we derive that which is a contradiction.
can be obtained from (4.4), which is inconsistent with , then there exists a sufficiently small > ε 0 0 such that for all ∈ r ε 0, 1 0 ( ], we have u r α r u r r r 0, and 0, 0, . Based on the fact that α α , u v ( ) is a lower solution of (2.1) and f 1 is quasi-monotone nondecreasing with respect to t, we integrate the first equation of problem (4.3) from 0 to r 1 and obtain ) ) It follows from Lemma 2.2 and In this section, we consider the non-existence, existence, and multiplicity of radial k-convex solutions for system (1.1).At first, we try to deal with the following system: Let ρ ( ) be the circle in 2 entered at the origin with radius ρ.

{( ) }
From this, we know that Θ is nonempty and unbounded in both directions of axes λ 0 1 and λ 0 2 .
Proof.i ( ) From Lemma 2.2, there exists a sufficiently large constant ρ such that problem (5.1) has at least one positive solution in ρ 0 ( ).Let ∈ u v ρ , 0 ( ) ( ) be a positive solution of (5.1).It follows from Theorem 2.8 that u and v are both strictly decreasing.
We denote ≔ L ν r max ).By the strictly decreasing property of u and v on R 0, [ ], we deduce that and From Lemma 2.1, we have for any . Combining this with > p q N , 1 2 , we know Now, we consider the two nonempty sets and define

3 Theorem 1 . 1 .
k-Convex solutions for k-Hessian systems  Assume that (H) holds.Then there exist

2 0
, problem (5.1) has a second positive solution.For this, let u v , () be the lower solution as we constructed before.
set.We reconsider problem (5.1) in the following form: