奇异k-Hessian方程的多个径向解
Many Radial Solutions of Singular k-Hessian Equations
摘要: 本文研究奇异k-Hessian方程多个非平凡径向解的存在性： 其中， 是k-Hessian算子，B表示RN (N≥2)中的单位球。研究的主要意义在于权函数 在B的边界上无界，并且证明上述k-Hessian有多个非平凡解。研究方法主要采用不动点指数定理。
Abstract: We prove that many nontrivial radial solutions exist for the singular k-Hessian problem Here is the k-Hessian operator, and B is the unit ball in RN (N≥2) . The main interest is that the weight function is unbounded as , and many nontrivial radial solutions to the above k-Hessian problem are derived. Our approach to show existence and multiplicity, exploits fixed point index theory.
文章引用：孙华远, 冯美强. 奇异k-Hessian方程的多个径向解[J]. 理论数学, 2018, 8(3): 289-295. https://doi.org/10.12677/PM.2018.83038

参考文献

[1] Trudinger, N. and Wang, X. (2008) The Monge-Ampère Equation and Its Geometric Applications. Handbook of Geometric Analysis, 1, 467-524. http://pdfs.semanticscholar.org/3e98/fded69b4c929d882393a609831df67a48e92.pdf
[2] Brandolini, B. (2013) On the Symmetry of Solutions to a k-Hessian Type Equation. Advanced Nonlinear Studies, 13, 487-493. [Google Scholar] [CrossRef] 
[3] Della Pietra, F. and Gavitone, N. (2014) Upper Bounds for the Eigenvalues of Hessian Equations. Annali di Matematica Pura ed Applicata, 193, 923-938. [Google Scholar] [CrossRef] 
[4] Gavitone, N. (2009) Isoperimetric Estimates for Eigenfunctions of Hessian Operators. Ricerche di Matematica, 58, 163-183. [Google Scholar] [CrossRef] 
[5] Zhang, X. and Feng, M. (2018) Boundary Blow-Up Solutions to the k-Hessian Equation with Singular Weights. Nonlinear Analysis, 167, 51-66. [Google Scholar] [CrossRef] 
[6] Sánchez, J. and Vergara, V. (2016) Bounded Solutions of a k-Hessian Equation in a Ball. Journal of Differential Equations, 261, 797-820. [Google Scholar] [CrossRef] 
[7] Nakamori, S. and Takimoto, K. (2015) A Ernstein Type Theorem for Parabolic k-Hessian Equations. Nonlinear Analysis, 117, 211-220. [Google Scholar] [CrossRef] 
[8] Wang, Q. and Xu, C.-J. (2014) C1;1 Solution of the Dirichlet Problem for Dege-nerate k-Hessian Equations. Nonlinear Analysis, 104, 133-146. [Google Scholar] [CrossRef] 
[9] Huang, Y. (2010) Boundary Asymptotical Behavior of Large Solutions to Hessian Equations. Pacific Journal of Mathematics, 244, 85-98. [Google Scholar] [CrossRef] 
[10] Dai, L. and Bao, J. (2011) On Uniqueness and Existence of Viscosity Solutions to Hessian Equations in Exterior Domains. Frontiers of Mathematics in China, 6, 221-230. [Google Scholar] [CrossRef] 
[11] Wang, C. and Bao, J. (2013) Necessary and Sufficient Conditions on Existence and Convexity of Solutions for Dirichlet Problems of Hessian Equations on Exterior Domains. Proceedings of the American Mathematical Society, 141, 1289-1296. [Google Scholar] [CrossRef] 
[12] Urbas, J.I.E. (1990) On the Existence of Nonclassical Solutions for Two Classes of Fully Nonlinear Elliptic Equations. Indiana University Mathematics Journal, 39, 355-382. [Google Scholar] [CrossRef] 
[13] Wang, X.-J. (1994) A Class of Fully Nonlinear Elliptic Equations and Related Functionals. Indiana University Mathematics Journal, 43, 25-54. [Google Scholar] [CrossRef] 
[14] Wang, X.-J. (2009) The k-Hessian Equation, in: Geometric Analysis and PDEs. Lecture Notes in Mathematics, 1977, 177-252. [Google Scholar] [CrossRef] 
[15] Clément, P., De Fi-gueiredo, D.G. and Mitidieri, E. (1996) Quasilinear Elliptic Equations with Critical Exponents. Topological Methods in Nonlinear Analysis, 7, 133-170. [Google Scholar] [CrossRef] 
[16] Wang, F. and An, Y. (2012) Triple Nontrivial Radial Convex Solutions of Systems of Monge-Ampère Equations. Applied Mathematics Letters, 25, 88-92. [Google Scholar] [CrossRef] 
[17] Wang, H. (2006) Convex Solutions of Boundary Value Problems. Journal of Mathematical Analysis and Applications, 318, 246-252. [Google Scholar] [CrossRef] 
[18] Han, X., Ma, R. and Dai, G. (2015) Eigenvalue, Bifurcation and Convex Solutions for Monge-Ampère Equations. Topological Methods in Nonlinear Analysis, 45, 135-164. [Google Scholar] [CrossRef] 
[19] Sánchez, J. and Vergara, V. (2017) Bounded Solutions of a k-Hessian Equation Involving a Weighted Nonlinear Source. Journal of Differential Equations, 263, 687-708. [Google Scholar] [CrossRef] 
[20] Jacobsen, J. (1999) Global Bifurcation Problems Associated with k-Hessian Op-erators. Topological Methods in Nonlinear Analysis, 14, 81-130. [Google Scholar] [CrossRef] 
[21] Escudero, C. and Torres, P.J. (2015) Existence of Radial Solutions to Biharmonic K-Hessian Equations. Journal of Differential Equations, 259, 2732-2761. [Google Scholar] [CrossRef] 
[22] Zhang, Z. and Zhou, Y. (2015) Existence of Entire Positive k-Convex Radial Solutions to Hessian Equations and Systems with Weights. Applied Mathematics Letters, 50, 48-55. [Google Scholar] [CrossRef] 
[23] Wei, W. (2016) Uniqueness Theorems for Negative Radial Solutions of k-Hessian Equations in a Ball. Journal of Differential Equations, 261, 3756-3771. [Google Scholar] [CrossRef] 
[24] Lan, K. (2001) Multive Positive Solutions of Semilinear Differential Equations with Singularities. Journal of the London Mathematical Society, 63, 690-704. [Google Scholar] [CrossRef] 

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