A class of singular $k_i$-Hessian systems
DOI:
https://doi.org/10.12775/TMNA.2022.072Keywords
Singular $k_i$-Hessian systems, positive solutions, topological methods, existence, nonexistence and multiplicityAbstract
Our main objective of this article is to investigate a class of singular $k_i$-Hessian systems. Among others, we obtain new theorems on the existence and multiplicity of positive radial solutions. Several nonexistence theorems are also derived.References
H. Amann, Fixed point equations and nonlinear eigenvalue problems in order Banach spaces, SIAM Rev. 18 (1976), 620–709.
J. Bao, H. Li and Y.Y. Li, On the exterior Dirichlet problem for Hessian equations, Trans. Amer. Math. Soc. 366 (2014), 6183–6200.
D.P. Covei, A remark on the existence of entire large and bounded solutions to a (k1 , k2 )Hessian system with gradient term, Acta Math. Sinica 33 (2017), 761–774.
D.P. Covei, A remark on the existence of positive radial solutions to a Hessian system, AIMS Math. 6 (2021), 14035–14043.
J. Cui, Existence and nonexistence of entire k-convex radial solutions to Hessian type system, Adv. Difference Equ. 2021 (2021), article no. 462.
G. Dai, Bifurcation and admissible solutions for the Hessian equation, J. Funct. Anal. 273 (2017), 3200–3240.
M. Feng, New results of coupled system of k-Hessian equations, Appl. Math. Lett. 94 (2019), 196–203.
M. Feng, Convex solutions of Monge–Ampére equations and systems: Existence, uniqueness and asymptotic behavior, Adv. Nonlinear Anal. 10 (2021), 371–399.
M. Feng and X. Zhang, A coupled system of k-Hessian equations, Math. Methods Appl. Sci. 44 (2021), 7377–7394.
C. Gao, X. He and M. Ran, On a power-type coupled system of k-Hessian equations, Quaest. Math. 44 (2021), 1593–1612.
B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 8 (1999), 45–69.
D.D. Hai, Existence and uniqueness of solutions for quasilinear elliptic systems, Proc. Amer. Math. Soc. 133 (2004), 223–228.
D.D. Hai and R. Shivaji, An existence result on positive solutions for a class of semilinear elliptic systems, Proc. Edinb. Math. Soc. Ser. A 134 (2004), 137–141.
D.D. Hai and H.Y. Wang, Nontrivial solutions for p-Laplacian systems, J. Math. Anal. Appl. 330 (2007), 186–194.
Y. Huang, Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math. 244 (2010), 85–98.
J. Ji, F. Jiang and B. Dong, On the solutions to weakly coupled system of ki -Hessian equations, J. Math. Anal. Appl. 513 (2022), 126217.
F. Jiang and N. Trudinger, On the Dirichlet problem for general augmented Hessian equations, J. Differential Equations 269 (2020), 5204–5227.
K.Q. Lan, Nonzero positive solutions of systems of elliptic boundary value problems, Proc. Amer. Math. Soc. 139 (2011), 4343–4349.
K.Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl. 394 (2012), 581–591.
D. Li and S. Ma, Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides, J. Funct. Anal. 276 (2019), 2969–2989
N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151–164.
J.I.E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J.39 (1990), 355–382.
F. Wang and Y. An, Triple nontrivial radial convex solutions of systems of Monge–Ampère equations, Appl. Math. Lett. 25 (2012), 88–92.
G. Wang, Z. Yang, L. Zhang and D. Baleanu, Radial solutions of a nonlinear kHessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91 (2020), 105396.
H. Wang, Convex solutions of systems arising from Monge–Ampère equations, Electron. J. Qual. Theory Differ. Equ. 26 (2009), Special Edition I, 1—8.
X.J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994), 25–54.
W. Wei, Uniqueness theorems for negative radial solutions of k-Hessian equations in a ball, J. Differential Equations 261 (2016), 3756–3771.
W. Wei, Existence and multiplicity for negative solutions of k-Hessian equations, J. Differential Equations 263 (2017), 615–640.
X. Zhang, Existence and uniqueness of nontrivial radial solutions for k-Hessian equations, J. Math. Anal. Appl. 492 (2020), 124439.
X. Zhang and Y. Du, Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equation, Calc. Var. Partial Differential Equations 57 (2018), article no. 30.
X. Zhang and M. Feng, The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation, J. Differential Equations 267 (2019), 4626–4672.
X. Zhang and M. Feng, Blow-up solutions to the Monge-Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates, Calc. Var. Partial Differerential Equations 61 (2022), article no. 208.
X. Zhang, L. Liu, Y. Wu and Y. Cui, A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator, Nonlinear Anal. Model. Control 25 (2020), 126–143.
X. Zhang, P. Xu and Y. Wu, The eigenvalue problem of a singular k-Hessian equation, Appl. Math. Lett. 124 (2022), 107666.
Z. Zhang and Z. Qi, On a power-type coupled system of Monge–Ampère equations. Topol. Methods Nonlilear Anal. 46 (2015), 717–729.
Z. Zhang and S. Zhou, Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights, Appl. Math. Lett. 50 (2015), 48–55.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Meiqiang Feng
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
