Abstract
In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.
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1 Introduction
In this paper, we study the existence of entire k-convex radial solutions to the following problem of Hessian type system:
where \(k,l=1,2,\ldots, N\), \(\mu , \nu \ge 0\) are constants, \(B_{1}(0)\) is the unit ball in \(\mathbb{R}^{N}\), for any \(N \times N\) real symmetric matrix A, \(\lambda (A)\) denotes the eigenvalues of A, \(D^{2} u(x)= ( \frac{\partial ^{2} u(x)}{\partial x_{i} \partial x_{j}} )\) denotes the Hessian matrix of the function \(u \in C^{2} (\overline{B_{1}(0)} )\), ∇u denotes the gradient of u, and \(\sigma _{k}(\lambda )=\sum_{1 \leq i_{1}<\cdots <i_{k} \leq N} \lambda _{i_{1}} \cdots \lambda _{i_{k}}\) denotes the kth elementary symmetric function of \(\lambda = (\lambda _{1},\ldots, \lambda _{N} ) \in \mathbb{R}^{N}\).
For p, q, \(f_{1}\), \(f_{2}\), \(g_{1}\), \(g_{2}\), we introduce the following conditions:
-
(H1)
\(p,q\in C([0,1],(0,+\infty ))\). \(f_{1},f_{2},g_{1},g_{2}\in C((-\infty ,0],[0,+\infty ))\) are decreasing.
-
(H2)
For any \(a>0\), the integral \(\int _{-\infty }^{-a} \frac{d\tau }{(f_{1}(\tau )f_{2}(\tau ))^{\frac{1}{k}}+(g_{1}(\tau )g_{2}(\tau ))^{\frac{1}{l}}}\) is divergent.
-
(H3)
For any \(a>0\), the integral \(\int _{-a}^{0} \frac{d\tau }{(f_{1}(\tau )f_{2}(\tau ))^{\frac{1}{k}}+(g_{1}(\tau )g_{2}(\tau ))^{\frac{1}{l}}}\) is divergent.
Denote
We say that a function \(u \in C^{2} (\overline{B_{1}(0)} )\) is k-convex in \(B_{1}(0)\) if \(\lambda (D^{2} u(x) ) \in \Gamma _{k}\) for all \(x \in B_{1}(0)\).
In (1.1), if \(\mu =0\) and \(f_{2}(v)\equiv 1\), the first equation in the system becomes the following k-Hessian type equation:
if \(\mu =\nu =0\) and \(f_{1}(u)=g_{2}(v)\equiv 1\), the system becomes the following coupling k-Hessian system:
Related to k-Hessian equations, if \(k=1\) the k-Hessian equations become the well-known Laplacian equations, and if \(k=N\) the k-Hessian equations become the Monge–Ampère equations. Concerning Laplacian equations and Monge–Ampère equations, there are a great number of research papers, see for examples [1, 6, 7, 22] and the references therein. Here we specially mention Keller [15], Osserman [21], and Lair and Wood [17] for Laplacian equations and Cheng and Yau [2] and Laser and McKenna [19] for Monge–Ampère equations. Similar situations occur for coupling k-Hessian system (1.3), although in this case there are not so many research papers. Here we only mention Lair and Wood [18] and Cîrstea and Rădulescu [3] for coupling Laplacian systems and Wang and An [24] and Zhang and Qi [26] for coupling Monge–Ampère systems.
For general k-Hessian equation (1.2), when \(p \equiv 1\) and \(f(u)=u^{\gamma k}\), \(\gamma >1\), Jin, Li, and Xu [13] showed the nonexistence of entire k-convex positive solutions. When \(p \equiv 1\), Ji and Bao [11] gave necessary and sufficient conditions on the existence of entire positive k-convex radial solutions. If we generalize \(p( \vert x \vert )f(u)\) to \(f(x,u)\), de Oliveira, do Ó, and Ubilla obtained the existence of k-convex radial solutions in the case of supercritical nonlinearity by means of variational techniques (see [5] and the references therein for research in this direction). For general k-Hessian equation (1.2) and coupling k-Hessian system (1.3), Zhang and Zhou [27] obtained several results on the existence of entire positive k-convex radial solutions. We refer to the papers of Feng and Zhang [8] and Gao, He, and Ran [9] and the references therein for research on coupling k-Hessian system (1.3).
It is obvious that the k-Hessian type equation
is a generalization of k-Hessian equation (1.2), but it is a special case of the following fully nonlinear Hessian equation:
See Guan and Jiao [10] and Jiang and Trudinger [12] and the references therein for research on fully nonlinear Hessian equation (1.4). Here we also want to mention the work of Dai [4] for similar study.
Inspired by the works above, and as we know that now there are no papers on the problem of k-Hessian type system (1.1), we obtain the following results in this paper.
Theorem 1.1
Under conditions (H1) and (H2), if \(f_{1}(0)g_{2}(0)\neq0\) and \(f_{2}(0)+g_{1}(0)\neq0\), then problem (1.1) admits an entire k-convex radial solution \((u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )\).
Remark 1.1
In the case of \(f_{1}(0)g_{2}(0)=0\), if \(f_{1}(0)=g_{2}(0)=0\), then there is a trivial solution \((u,v)=(0,0)\) to problem (1.1); if \(f_{1}(0)=0\) or \(g_{2}(0)=0\), then there is a semi-trivial solution \((u,v)=(0,v)\) or \((u,v)=(u,0)\) to problem (1.1); moreover, the semi-trivial solution may become trivial if \(f_{1}(0)=0\) with \(g_{1}(0)=0\) or \(g_{2}(0)=0\) with \(f_{2}(0)=0\).
In the case of \(f_{2}(0)+g_{1}(0)=0\), there is a trivial solution \((u,v)=(0,0)\) to problem (1.1).
Theorem 1.2
Under conditions (H1) and (H3), problem (1.1) admits no entire k-convex radial solution \((u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )\).
Remark 1.2
In this case, \(f_{1}(0)f_{2}(0)=g_{1}(0)g_{2}(0)=0\), and there is a trivial solution \((u,v)=(0,0)\) to problem (1.1).
2 Preliminaries
In this section, we give some preliminary results which will be used to prove the main results in the next section.
Lemma 2.1
Assume \(\varphi (r) \in C^{2}[0, 1]\) with \(\varphi ^{\prime }(0)=0\). Then, for \(u(x)=\varphi (r)\), there holds that \(u \in C^{2} (\overline{B_{1}(0)} )\) and
and further
where \(C_{N}^{k}=\frac{N!}{k!(N-k)!}\).
Proof
It is immediate that, for \(x \neq 0\), \(1 \leq i, j \leq N\),
and
Further if define
then \(u \in C^{2} (\overline{B_{1}(0)} )\).
Now it is easy to show the two equalities for \(\lambda (D^{2} u+\eta \vert \nabla u \vert I )\) and \(\sigma _{k} (\lambda (D^{2} u+\eta \vert \nabla u \vert I ) )\). □
Lemma 2.2
Let \(f\in C(-\infty ,0]\) be decreasing. Assume that \(\varphi \in C^{0}[0, 1] \cap C^{1}(0, 1]\) is a solution of the Cauchy problem
where
Then \(\varphi \in C^{2}[0, 1]\), and it satisfies the problem
Furthermore, if φ is nontrivial, i.e., \(\varphi (r)<0\) for \(0\le r<1\), then
for \(0 \leq r<1\).
Proof
It is easy to see that \(\varphi (r) \in C^{2}[0, 1]\).
From
we have
and further differentiating with respect to r we have
If φ is nontrivial, it is easy to see that φ is increasing, so for \(0 \leq r<1\) we conclude \(\varphi (r)<\varphi (1)=0\), \(f(\varphi (r))>f(\varphi (1))\ge 0\) and further
By the properties of kth elementary symmetric functions (see for example [20]), we know \(\sigma _{j} (\lambda _{r} )>0\) for \(1\le j< k\) and \(0 \leq r<1\). Therefore we conclude the lemma. □
3 Proofs of the main results
In this section, we prove the main results in this paper, i.e., the existence and nonexistence of entire k-convex radial solutions for problem (1.1).
Proof of Theorem 1.1
From the system
we get
furthermore we have
Define
then we need only to find a fixed point of \(\mathcal{L}\). Here we use the monotone iterative method to find such a fixed point.
It is easy to show that \(\mathcal{L}\) is a mapping from \(C^{2}[0,1]\times C^{2}[0,1]\) to \(C^{2}[0,1]\times C^{2}[0,1]\), and it is continuous on \(C[0,1]\times C[0,1]\).
Let \(\{u_{n}\}\) and \(\{v_{n}\}\) be the sequence of continuous functions defined by
It is easy to see that \(u_{n}\) and \(v_{n}\) are decreasing on \([0,1]\) for \(n>1\) and by induction \(\{u_{n}\}\) and \(\{v_{n}\}\) are decreasing as well, i.e., \(u_{n+1}(r)< u_{n}(r)\) and \(v_{n+1}(r)< v_{n}(r)\) for \(0\leq r<1\) and \(n\ge 1\).
By condition (H1), for each \(0< r<1\) and \(n>1\),
where \(C(N,k,p)\) is a constant dependent on N, k, and p.
Similarly,
and further
i.e.,
where \(C(N,l,q)\) and \(C(N,k,l,p,q)\) are constants dependent on N, l, q and N, k, l, p, q, respectively.
Integrating from 1 to r, we have
By condition (H2), denote
then F is continuous and increasing on \((-\infty ,0]\), and it has an inverse function \(F^{-1}\). From (3.2), we have
for \(0\le r\le 1\) and \(n\ge 1\).
By condition (H1) and (3.1), we have for \(n\ge 1\)
where \(C(N,k,l,p,q, f_{1},f_{2},g_{1},g_{2})\) is a constant dependent on N, k, l, p, q, \(f_{1}\), \(f_{2}\), \(g_{1}\), and \(g_{2}\). So \(\{u_{n}\}\) and \(\{v_{n}\}\) are bounded in \(C^{1}[0,1]\) and by Arzela–Ascoli theorem \(\{u_{n}\}\) and \(\{v_{n}\}\) have convergent subsequences (still denoted by \(\{u_{n}\}\) and \(\{v_{n}\}\)) in \(C[0,1]\). Denote
By the continuity of \(\mathcal{L}\) on \(C[0,1]\times C[0,1]\), from
we conclude that \((u,v)\) is a fixed point of \(\mathcal{L}\) after letting \(n\rightarrow +\infty \). □
Proof of Theorem 1.2
We prove by contradiction. Suppose that \((u,v)\) is a k-convex radial solution to problem (1.1). Then u and v are decreasing on \([0,1]\). For \(0< r< 1\), by Lemma 2.2 we can get
So
Integrating for 0 to 1, we have
which contradicts condition (H3). Now we finish the proof. □
At the end of this section, we give some examples for the sake of clearly understanding the results in this paper.
Assume that α, β, \(\alpha _{1}\), \(\beta _{1}\), \(\alpha _{2}\), and \(\beta _{2}\) are positive.
Example 3.1
If \(\alpha _{1} +\beta _{1}\le k\) and \(\alpha _{2} +\beta _{2}\le l\), then the following problem admits an entire k-convex radial solution \((u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )\):
Example 3.2
If \(\alpha _{1} +\beta _{1}\ge k\) and \(\alpha _{2} +\beta _{2}\ge l\), then the following problem admits no entire k-convex radial solution \((u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )\):
4 Conclusion
In this paper, by converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, we establish the existence of entire k-convex radial solutions for a Hessian type system. Moreover the nonexistence of entire k-convex radial solutions is also obtained. In the process of obtaining the existence of entire k-convex radial solutions, we utilize the monotone iterative method. By different fixed point theorems (such as the ones in [14] and [25]) or different methods (such as degree theory in [16] and the regularization method in [23]), we may get different results on Hessian type systems. In our opinion, it is interesting to fulfil this kind of works in the future.
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References
Cavalheiro, A.C.: Existence results for Navier problems with degenerated \((p,q)\)-Laplacian and \((p,q)\)-biharmonic operators. Results Nonlinear Anal. 1(2), 74–87 (2018)
Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33, 507–544 (1980)
Cîrstea, F.C., Rădulescu, V.: Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl. 81, 827–846 (2002)
Dai, L.M.: Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete Contin. Dyn. Syst. 40(1), 579–596 (2020)
de Oliveira, J.F., do Ó, J.M., Ubilla, P.: Existence for a k-Hessian equation involving supercritical growth. J. Differ. Equ. 267, 1001–1024 (2019)
Enache, C., Porru, G.: A note on Monge–Ampere equation in \(\mathbb{R}^{2}\). Results Math. 76(1), Article ID 29 (2021)
Feng, M.Q.: Convex solutions of Monge–Ampère equations and systems: existence, uniqueness and asymptotic behavior. Adv. Nonlinear Anal. 10(1), 371–399 (2021)
Feng, M.Q., Zhang, X.M.: A coupled system of k-Hessian equations. Math. Methods Appl. Sci. 44(9), 7377–7394 (2021)
Gao, C.H., He, X.Y., Ran, M.J.: On a power-type coupled system of k-Hessian equations. Quaest. Math. https://doi.org/10.2989/16073606.2020.1816586
Guan, B., Jiao, H.: Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 54, 2693–2712 (2015)
Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138, 175–188 (2010)
Jiang, F., Trudinger, N.S.: On the Dirichlet problem for general augmented Hessian equations. J. Differ. Equ. 269, 5204–5227 (2020)
Jin, Q., Li, Y., Xu, H.: Nonexistence of positive solutions for some fully nonlinear elliptic equations. Methods Appl. Anal. 12, 441–450 (2005)
Karapınar, E.: A fixed point theorem without a Picard operator. Results Nonlinear Anal. 4(3), 127–129 (2021)
Keller, J.B.: On solutions of \(\triangle u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)
Kim, I.S.: Semilinear problems involving nonlinear operators of monotone type. Results Nonlinear Anal. 2(1), 25–35 (2019)
Lair, A.V., Wood, A.W.: Large solutions of semilinear elliptic problems. Nonlinear Anal. 37, 805–812 (1999)
Lair, A.V., Wood, A.W.: Existence of entire large positive solutions of semilinear elliptic systems. J. Differ. Equ. 164, 380–394 (2000)
Lazer, A.C., McKenna, P.J.: On singular boundary value problems for the Monge–Ampère operator. J. Math. Anal. Appl. 197, 341–362 (1996)
Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, New Jersey (1996)
Osserman, R.: On the inequality \(\triangle u \geq f(u)\). Pac. J. Math. 7, 1641–1647 (1957)
Ourraou, A.: Existence and uniqueness of solutions for Steklov problem with variable exponent. Adv. Theory Nonlinear Anal. Appl. 5(1), 158–166 (2021)
Phuong, N.D., Luc, N.H., Long, L.D.: Modified quasi boundary value method for inverse source problem of the bi-parabolic equation. Adv. Theory Nonlinear Anal. Appl. 4(3), 132–142 (2020)
Wang, F., An, Y.: Triple nontrivial radial convex solutions of systems of Monge–Ampère equations. Appl. Math. Lett. 25, 88–92 (2012)
Zhang, C., Chen, J.: Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. Results Nonlinear Anal. 2(3), 102–112 (2019)
Zhang, Z., Qi, Z.: On a power-type coupled system of Monge–Ampère equations. Topol. Methods Nonlinear Anal. 46, 717–729 (2015)
Zhang, Z., Zhou, S.: Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Appl. Math. Lett. 50, 48–55 (2015)
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The author is grateful to the referees for their helpful remarks and suggestions.
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I am supported by the National Natural Science Foundation of China (Grant No. U2031142).
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Cui, J. Existence and nonexistence of entire k-convex radial solutions to Hessian type system. Adv Differ Equ 2021, 462 (2021). https://doi.org/10.1186/s13662-021-03601-8
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DOI: https://doi.org/10.1186/s13662-021-03601-8