The existence and multiplicity of nontrivial solutions for a class of p-Monge-Ampère system
DOI:
https://doi.org/10.12775/TMNA.2025.043Keywords
p-Monge-Ampère equation, radial solutions, existence, multiplicity, fixed-point theoremAbstract
In this paper, we deal with the following p-Monge-Ampère system: \begin{equation*} \begin{cases} \text{det}(D(|Du_{1}|^{p-2}Du_{1}))=f_{1}(|x|,-u_{2}), & x\in B,\\ \text{det}(D(|Du_{2}|^{p-2}Du_{2}))=f_{2}(|x|,-u_{1}),& x\in B,\\ u_{1}=u_{2}=0, & x\in\partial B, \end{cases} \end{equation*} where $B=\{x\in\mathbb{R}^{n}:|x|< 1\}$ and $f_{i}$ $(i=1,2)$ are continuous and nonnegative functions. Based on the fixed-point theory, some results regarding existence of radial solutions are established when $f_{i}$ $(i=1,2)$ satisfy some new growth conditions.References
T. Alotaibi, D.D. Hai and R. Shivaji, Existence and nonexistence of positive radial solutions for a class of p-Laplacian superlinear problems with nonlinear boundary conditions, Commun. Pure Appl. Anal. 19 (2020), 4655–4666.
L. An, On the local Hölder continuity of the inverse of the p-Laplace operator, Proc. Amer. Math. Soc. 135 (2007), 3553–3560.
I. Bakelman, Convex Analysis and Nonlinear Geometric Equations, Proc. Amer. Math. Soc., Springer, Berlin, 1994.
K.D. Chu, D.D. Hai and R. Shivaji, Uniqueness of positive radial solutions for infinite semipositone p-Laplacian problems in exterior domains, J. Math. Anal. Appl. 472 (2019), 510–525.
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
M.Q. Feng, Eigenvalue problems for singular p-Monge–Ampère equations, J. Math. Anal. Appl. 528 (2023), 17 pp.
M.Q. Feng, Nontrivial p-convex solutions to singular p-Monge–Ampere problems: existence, multiplicity and nonexistence, Commun. Anal. Mech. 16 (2024), 71–93.
C.H. Gao, X.Y. He and J.J. Wang, The existence and multiplicity of k-convex solutions for a coupled k-Hessian system, Acta Math. Sci. Ser. B (Engl. ed.) 43 (2023), 2615–2628.
C.H. Gao, X.Y. He and M.G. Ran, On a power-type coupled system of k-Hessian equations, Quaest. Math. 44 (2021), 1593–1612.
X.M. He and W.G. Ge, Twin positive solutions for the one-dimensional p-Laplacianboundary value problems, Nonlinear Anal. 56 (2004), 975–984.
J.K. Liu, A class of nonlinear optimisation and applications, Methods Appl. Anal. 28 (2021), 31–52.
R.H. Liu, F.L. Wang and Y.K. An, On radial solutions for Monge–Ampère equations, Turkish J. Math. 42 (2018), 1590–1609.
K.Q. Lan and Z.T. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl. 394 (2012), 581–591.
X. Ma, N. Trudinger and X. Wang, Regularity of potential functions of the optential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005), 151–183.
R. Shivaji, I. Sim and B. Son, A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl. 445 (2017), 459–475.
N. Trudinger and X. Wang, Hessian measures II, Ann. of Math. 150 (1999), 579–604.
J.Y. Wang, The existence of positive solutions for the one-dimensional p-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 2275–2283.
Z.T. Zhang and S.J. Li, On sign-changing and multiple solutions of the p-Laplacian, J. Funct. Anal. 197 (2003), 447–468.
Z.T. Zhang and Z.X. Qi, On a power-type coupled system of Monge–Ampère equations, Topol. Methods Nonlinear Anal. 46 (2015), 717–729.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
