In this paper, we are concerned with the existence of the positive bounded and blow-up radial solutions of the $(k_{1},k_{2})$-Hessian system \begin{equation*} \begin{split} \left\{\begin{array}{l}{\mathcal{G} (K_{1}^{\frac{1}{k_{1}}}) K_{1}^{\frac{1}{k_{1}}}=b_{1}(|x|) g_{1}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}}, \\ {\mathcal{G}(K_{2}^{\frac{1}{k_{2}}}) K_{2}^{\frac{1}{k_{2}}}=b_{2}(|x|) g_{2}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}},\end{array}\right. \end{split} \end{equation*} where $\mathcal{G}$ is a nonlinear operator, $K_{i}=S_{k_{i}}\left(\lambda\left(D^{2} z_{i}\right)\right)+\psi_{i}(|x|)|\nabla z_{i}|^{k_{i}},i=1,2.$ Under the appropriate conditions on $g_{1}$ and $g_{2}$, our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem. Furthermore, our main results also extend the previous existence results for both the single equation and systems.
Guotao WANG
,
Zedong YANG
,
Jiafa XU
,
Lihong ZHANG
. THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT[J]. Acta mathematica scientia, Series B, 2022
, 42(4)
: 1414
-1426
.
DOI: 10.1007/s10473-022-0409-0
[1] Ghergu M, Rădulescu V, Explosive solutions of semilinear elliptic systems with gradient term. Rev R Acad Cienc Exactas Fís Nat Ser A Mat, 2003, 97:437-445
[2] García-Melián J, A remark on uniqueness of large solutions for elliptic systems of competitive type. J Math Anal Appl, 2007, 331:608-616
[3] Lair A V, Mohammed A, Large solutions to semi-linear elliptic systems with variable exponents. J Math Anal Appl, 2014, 420:1478-1499
[4] Covei D, Solutions with radial symmetry for a semilinear elliptic system with weights. Appl Math Lett, 2018, 76:187-194
[5] Li H, Zhang P, Zhang Z, A remark on the existence of entire positve solutions for a class of semilinear elliptic system. J Math Anal Appl, 2010, 365:338-341
[6] Yang H, Chang Y, On the blow-up boundary solutions of the Monge-Ampère equation with singular weights. Comm Pure Appl Anal, 2012, 11:697-708
[7] Lair A V, A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems. J Math Anal Appl, 2010, 365:103-108
[8] Guan B, Jian H, The Monge-Ampère equation with infinite boundary value. Paciflc J Math, 2004, 216:77-94
[9] Zhang Z, Boundary behavior of large solutions to the Monge-Ampère equations with weights. J Differential Equations, 2015, 259:2080-2100
[10] Zhang X, Du Y, Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation. Calc Var Partial Differential Equations, 2018, 57:30
[11] Zhang Z, Large solutions to the Monge-Ampère equations with nonlinear gradient terms:existence and boundary behavior. J Differential Equations, 2018, 264:263-296
[12] Bao J, Li H, Zhang L, Monge-Ampère equation on exterior domains. Calc Var Partial Differential Equations, 2015, 52:39-63
[13] Zhang W, Bao J, A Calabi theorem for solutions to the parabolic Monge-Ampère equation with periodic data. Ann Inst H Poincaré Anal Non Linéaire, 2018, 35:1143-1173
[14] Zhang Z, Zhou S, Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Appl Math Lett, 2015, 50:48-55
[15] Covei D. A remark on the existence of entire large and bounded solutions to a (k1, k2)-Hessian system with gradient term. Acta Math Sin (Engl Ser), 2017, 33(6):761-774
[16] Wang G, Yang Z, Zhang L, Baleanu D, Radial solutions of a nonlinear k-Hessian system involving a nonlinear operator. Commun Nonlinear Sci Numer Simulat, 2020, 91:105396
[17] Zhang X, Wu Y, Cui Y, Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator. Appl Math Lett, 2018, 82:85-91
[18] Ji X, Bao J, Necessary and sufficient conditions on solvability for Hessian inequalities. Proc Amer Math Soc, 2010, 138:175-188
[19] Feng M, New results of coupled system of k-Hessian equations. Appl Math Lett, 2019, 94:196-203
[20] Balodis P, Escudero C, Polyharmonic k-Hessian equations in $\mathbb{R}$N. J Differential Equations, 2018, 265:3363-3399
[21] Zhang X, On a singular k-Hessian equation. Appl Math Lett, 2019, 97:60-66
[22] Zhang X, Feng M, The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation. J Differential Equations, 2019, 267:4626-4672
[23] Wang B, Bao J, Over-determined problems for k-Hessian equations in ring-shaped domains. Nonlinear Anal, 2015, 127:143-156
[24] Wang B, Bao J. Mirror symmetry for a Hessian over-determined problem and its generalization. Commun Pure Appl Anal, 2014, 13(6):2305-2316
[25] Wang G, Pei K, Agarwal R P, Zhang L, Ahmad B, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J Comput Appl Math, 2018, 343:230-239
[26] Zhang L, Ahmad B, Wang G, Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions. Appl Math Comput, 2015, 268:388-392
[27] Pei K, Wang G, Sun Y, Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain. Appl Math Comput, 2017, 312:158-168
[28] Wang G, Twin iterative positive solutions of fractional q-difference Schrödinger equations. Appl Math Lett, 2018, 76:103-109
[29] Wang G, Bai Z, Zhang L. Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem. J Appl Anal Comput, 2019, 9(4):1204-1215
