In this paper, by virtue of fixed point theory, we investigate a $ p $-Monge-Ampère problem and establish several existence results for positive radial solutions when the nonlinearity satisfies some $ (p-1)n $-superlinear and $ (p-1)n $-sublinear conditions.
Citation: Keyu Zhang, Houyu Zhao. Positive radial solutions for a $ p $-Monge-Ampère problem[J]. Mathematical Modelling and Control, 2025, 5(4): 390-399. doi: 10.3934/mmc.2025027
In this paper, by virtue of fixed point theory, we investigate a $ p $-Monge-Ampère problem and establish several existence results for positive radial solutions when the nonlinearity satisfies some $ (p-1)n $-superlinear and $ (p-1)n $-sublinear conditions.
| [1] |
J. Bao, Q. Feng, Necessary and sufficient conditions on global solvability for the $p$-$k$-Hessian inequalities, Canad. Math. Bull., 65 (2022), 1004–1019. https://doi.org/10.4153/s0008439522000066 doi: 10.4153/s0008439522000066
|
| [2] |
M. Feng, Nontrivial $p$-convex solutions to singular $p$-Monge-Ampère problems: existence, multiplicity and nonexistence, Commun. Anal. Mech., 16 (2024), 71–93. https://doi.org/10.3934/cam.2024004 doi: 10.3934/cam.2024004
|
| [3] | N. S. Trudinger, X. Wang, Hessian measures II, Ann. Math., 150 (1999), 579–604. |
| [4] |
Z. Bai, Z. Yang, Existence of $k$-convex solutions for the $k$-Hessian equation, Mediterr. J. Math., 20 (2023), 150. https://doi.org/10.1007/s00009-023-02364-8 doi: 10.1007/s00009-023-02364-8
|
| [5] |
W. Cao, L. Székelyhidi, Very weak solutions to the two-dimensional Monge-Ampère equation, Sci. China Math., 62 (2019), 1041–1056. https://doi.org/10.1007/s11425-018-9516-7 doi: 10.1007/s11425-018-9516-7
|
| [6] |
G. Du, X. Wang, Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator, Adv. Nonlinear Anal., 13 (2024), 20230135. https://doi.org/10.1515/anona-2023-0135 doi: 10.1515/anona-2023-0135
|
| [7] |
B. Emamizadeh, Y. Liu, G. Porru, Overdetermined problems for $p$-Laplace and generalized Monge-Ampère equations, Complex Var. Elliptic Equ., 67 (2022), 807–821. https://doi.org/10.1080/17476933.2020.1843447 doi: 10.1080/17476933.2020.1843447
|
| [8] |
M. Feng, A class of singular coupled systems of superlinear Monge-Ampère equations, Acta Math. Appl. Sin. Engl. Ser., 38 (2022), 925–942. https://doi.org/10.1007/s10255-022-1024-5 doi: 10.1007/s10255-022-1024-5
|
| [9] |
M. Feng, Convex solutions of Monge-Ampère equations and systems: existence, uniqueness and asymptotic behavior, Adv. Nonlinear Anal., 10 (2021), 371–399. https://doi.org/10.1515/anona-2020-0139 doi: 10.1515/anona-2020-0139
|
| [10] |
M. Feng, A class of singular $k_i$-Hessian systems, Topol. Methods Nonlinear Anal., 62 (2023), 341–365. https://doi.org/10.12775/tmna.2022.072 doi: 10.12775/tmna.2022.072
|
| [11] |
M. Feng, Eigenvalue problems for singular $p$-Monge-Ampère equations, J. Math. Anal. Appl., 528 (2023), 127538. https://doi.org/10.1016/j.jmaa.2023.127538 doi: 10.1016/j.jmaa.2023.127538
|
| [12] |
H. Gao, L. Wang, J. Li, Existence of radial solutions for $k$-Hessian system, AIMS Math., 8 (2023), 26498–26514. https://doi.org/10.3934/math.20231353 doi: 10.3934/math.20231353
|
| [13] |
N. Q. Le, Uniqueness for a system of Monge-Ampère equations, Methods Appl. Anal., 28 (2021), 15–30. https://doi.org/10.4310/MAA.2021.v28.n1.a2 doi: 10.4310/MAA.2021.v28.n1.a2
|
| [14] |
J. Shi, F. Jiang, On Neumann problem for the degenerate Monge-Ampère type equations, Bound. Value Probl., 2021 (2021), 11. https://doi.org/10.1186/s13661-021-01486-w doi: 10.1186/s13661-021-01486-w
|
| [15] |
M. Sroka, Weak solutions to the quaternionic Monge-Ampère equation, Anal. PDE, 13 (2020), 1755–1776. https://doi.org/10.2140/apde.2020.13.1755 doi: 10.2140/apde.2020.13.1755
|
| [16] |
Z. Yang, Z. Bai, Existence and multiplicity of radial solutions for a $k$-Hessian system, J. Math. Anal. Appl., 512 (2022), 126159. https://doi.org/10.1016/j.jmaa.2022.126159 doi: 10.1016/j.jmaa.2022.126159
|
| [17] |
Z. Yang, Z. Bai, Existence results for the $k$-Hessian type system with the gradients via $\Bbb R_+^n$-monotone matrices, Nonlinear Anal., 240 (2024), 113457. https://doi.org/10.1016/j.na.2023.113457 doi: 10.1016/j.na.2023.113457
|
| [18] |
Z. Yu, Z. Bai, The existence of radial solutions for a class of $k$-Hessian systems with the nonlinear gradient terms, J. Appl. Math. Comput., 70 (2024), 2225–2240. https://doi.org/10.1007/s12190-024-02049-9 doi: 10.1007/s12190-024-02049-9
|
| [19] |
X. Zhang, M. Feng, The existence and asymptotic behavior of boundary blow-up solutions to the $k$-Hessian equation, J. Differ. Equations, 267 (2019), 4626–4672. https://doi.org/10.1016/j.jde.2019.05.004 doi: 10.1016/j.jde.2019.05.004
|
| [20] |
X. Zhang, M. Feng, Boundary blow-up solutions to singular $k$-Hessian equations with gradient terms: sufficient and necessary conditions and asymptotic behavior, J. Differ. Equations, 375 (2023), 475–513. https://doi.org/10.1016/j.jde.2023.08.022 doi: 10.1016/j.jde.2023.08.022
|
| [21] | D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988. https://doi.org/10.1016/C2013-0-10750-7 |
| [22] |
J. Xu, Z. Yang, Positive solutions for a fourth order $p$-laplacian boundary value problem, Nonlinear Anal., 74 (2011), 2612–2623. https://doi.org/10.1016/j.na.2010.12.016 doi: 10.1016/j.na.2010.12.016
|
| [23] |
R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673–688. https://doi.org/10.1512/iumj.1979.28.28046 doi: 10.1512/iumj.1979.28.28046
|
Keyu Zhang, Houyu Zhao. Positive radial solutions for a $ p $-Monge-Ampère problem[J]. Mathematical Modelling and Control, 2025, 5(4): 390-399. doi: 10.3934/mmc.2025027
DownLoad: