Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
-
Zhiqian He
und
Liangying Miao
Abstract
This paper investigates the existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for the k-Hessian equation subject to a homogeneous Dirichlet boundary condition:
where B = {x ∈ ℝn: ∣x∣ < 1}, λ is a positive parameter, k ∈ {1, …, n}, f: [0, ∞) → [0, ∞) is continuous. In contrast to previous studies, our main results are established under the essential condition that the radially symmetric k-admissible solutions of (P) are generically non-convex. The main tool is the fixed point theorem in cones.
Funding statement: This work was supported by the National Natural Science Foundation of China (Nos.12461035, 12301631), Applied Basic Research Project of Qinghai Province (2025-ZJ-722).
Acknowledgement
The authors are very grateful to the anonymous referees for their valuable suggestions.
-
(Communicated by Michal Fečkan)
References
[1] Andrews, B.: Gauss curvature flow: The fate of the rolling stones, Invent. Math. 138 (1999), 151–161.10.1007/s002220050344Suche in Google Scholar
[2] Bai, Z. B.—Yang, Z. D.: Existence of k-convex solutions for the k-Hessian equation, Mediterr. J. Math. 20 (2023), Art. No. 150.10.1007/s00009-023-02364-8Suche in Google Scholar
[3] Caffarelli, L. A.—Milman, M.: Monge-Ampère equation: Applications to Geometry and Optimization, Amer. Math. Soc., Providence, RI, 1999.10.1090/conm/226Suche in Google Scholar
[4] Dai, G. W.—Ma, R. Y.: Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations, Topol. Methods Nonlinear Anal. 46 (2015), 135–163.10.12775/TMNA.2015.041Suche in Google Scholar
[5] Dai, G. W.: Bifurcation and admissible solutions for the Hessian equation, J. Funct. Anal. 273 (2017), 3200–3240.10.1016/j.jfa.2017.08.001Suche in Google Scholar
[6] Deimling, K.: Nonlinear Functional Analysis, Springer, Berlin, 1985.10.1007/978-3-662-00547-7Suche in Google Scholar
[7] Erbe, L. H.—Hu, S. C.—Wang, H. Y.: Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), 640–648.10.1006/jmaa.1994.1227Suche in Google Scholar
[8] Guan, P.—Wang, X.J.: On a Monge-Ampère equation arising in geometric optics, J. Differential Geometry 48 (1998), 205–223.10.4310/jdg/1214460795Suche in Google Scholar
[9] Hassine, K.: Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators, Commun. Pure Appl. Anal. 21 (2022), 2965–2979.10.3934/cpaa.2022084Suche in Google Scholar
[10] He, X. Y.—Gao, C. H.—Wang, J. J.: k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities, Adv. Nonlinear Anal. 13 (2024), Art. ID 20230136.10.1515/anona-2023-0136Suche in Google Scholar
[11] He, Z. Q.—Miao, L. Y.: Existence and multiplicity of radially symmetric k-admissible solutions for Dirichlet problem of k-Hessian equations, Math. Slovaca 72 (2022), 111–120.10.1515/ms-2022-0008Suche in Google Scholar
[12] Hu, S. C.—Wang, H. Y.: Convex Solutions of boundary value problems arising from Monge-Ampère equation, Discret. Contin. Dyn. Syst. 16 (2006), 705–720.10.3934/dcds.2006.16.705Suche in Google Scholar
[13] Jacobsen, J.: A Liouville-Gelfand equation for k-Hessian operators, Rocky Mountain J. Math. 34 (2004), 665–683.10.1216/rmjm/1181069873Suche in Google Scholar
[14] Ma, R. Y.—Gao, H. L.: Positive convex solutions of boundary value problems arising from Monge-Ampère equations, Appl. Math. Comput. 259 (2015), 390–402.10.1016/j.amc.2015.03.005Suche in Google Scholar
[15] Ma, R. Y.—He, Z. Q.—Yan, D. L.: Three radially symmetric k-admissible solutions for k-Hessian equation, Complex Var. Elliptic Equ. 64 (2019), 1353–1363.10.1080/17476933.2018.1536706Suche in Google Scholar
[16] Sánchez, J.—Vergara, V.: Bounded solutions of a k-Hessian equation in a ball, J. Differential Equations 261 (2016), 797–820.10.1016/j.jde.2016.03.021Suche in Google Scholar
[17] Wang, H. Y.: Convex solutions of boundary value problems, J. Math. Anal. Appl. 318 (2006), 246–252.10.1016/j.jmaa.2005.05.067Suche in Google Scholar
[18] Wei, W.: Uniqueness theorems for negative radial solutions of k-Hessian equations in a ball, J. Differential Equations 261 (2016), 3756–3771.10.1016/j.jde.2016.06.004Suche in Google Scholar
[19] Wei, W.: Existence and multiplicity for negative solutions of k-Hessian equations, J. Differential Equations 263 (2017), 615–640.10.1016/j.jde.2017.02.049Suche in Google Scholar
[20] Wang, X. J.: A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994), 25–54.10.1512/iumj.1994.43.43002Suche in Google Scholar
[21] Zhang, X. G.—Chen, P.—Wu, Y. H.—Wiwatanapataphee, B.: The iterative properties of solutions for a singular k-Hessian system, Nonlinear Anal. Model. Control 29 (2024), 146–165.10.15388/namc.2024.24.33824Suche in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
Artikel in diesem Heft
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
