一类Monge-Ampère系统非平凡径向凸解的存在性
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摘要
对一类由n个方程组成的Monge-Ampère系统,证明其非线性项为一般函数时该Monge-Ampère系统解的存在性。首先,在径向解的支撑下,通过一个巧妙的变换将Monge-Ampère系统转化为一个与之等价常微分方程系统;其次,在适当的Banach空间中,构造相应的非负锥和全连续算子;最后,利用锥上的不动点指数理论,在单位球内研究常微分方程系统正解的存在性。进一步得到了原Monge-Ampère系统非平凡径向凸解的存在性,并证明了在非线性项为超线性或次线性情况下,原Monge-Ampère系统至少存在一个非平凡径向凸解。
In this paper, we investigate the solution of a kind of Monge-Ampère systems composed of n equations. The existence of nontrivial radial convex solution for Monge-Ampère systems with general nonlinear terms is obtained. Firstly, the Monge-Ampère system can be transformed into an equivalent ordinary differential equation system by an ingenious transformation under the support of the radial solution. Secondly, we construct suitable nonnegative cone and completely continuous operators in Banach space. Finally, the existence of positive solutions of ordinary differential equations in unit sphere is studied by using the index theory of fixed points. And then the existence of nontrivial radial convex solutions of the original Monge-Ampère system can be obtained. We can prove that there is at least one nontrivial radial convex solution of the original Monge-Ampère system when the nonlinear term is superlinear or sublinear.
In this paper, we investigate the solution of a kind of Monge-Ampère systems composed of n equations. The existence of nontrivial radial convex solution for Monge-Ampère systems with general nonlinear terms is obtained. Firstly, the Monge-Ampère system can be transformed into an equivalent ordinary differential equation system by an ingenious transformation under the support of the radial solution. Secondly, we construct suitable nonnegative cone and completely continuous operators in Banach space. Finally, the existence of positive solutions of ordinary differential equations in unit sphere is studied by using the index theory of fixed points. And then the existence of nontrivial radial convex solutions of the original Monge-Ampère system can be obtained. We can prove that there is at least one nontrivial radial convex solution of the original Monge-Ampère system when the nonlinear term is superlinear or sublinear.
引文
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[ 7 ]WANG F,AN Y.Triple nontrivial radial convex solutions of systems of Monge-Ampère equations[J].Applied Mathematics Letters,2012,25:88-92.
[ 8 ]WANG H.Convex solutions of systems of Monge-Ampère equations[J].Journal of Mathematical Analysis and Applications,2006,318:246-251.
[ 9 ]GUTIERREZ C.The Monge-Ampère Equation[M].Basel:Birkhuser Publishing Ltd,2000.
[10]郭大均.非线性泛函分析[M].3版.济南:山东科学技术出版社,2015.
[11]冯美强,张学梅.Banach空间中非线性常微分方程边值问题[M].北京:科学出版社,2018.
[ 2 ]WANG H.Raidal convex solutions of boundary value problems for systems of Monge-Ampère equations[J].Analysis & Partial Differential Equations,2010.
[ 3 ]WANG H.Convex solutions of systems arising from Monge-Ampère equations[J].Electronic Journal Qualitative Theory of Differential Equations,2009,26:1-8.
[ 4 ]ZHANG Z,WANG K.Existence and non-existence of solutions for a class of Monge-Ampère equations[J].Differential Equations,2009,246:2849-2875.
[ 5 ]KUTEV N.Nontrivial solutions for the equations of Monge-Ampere type[J].Journal of Mathematical Analysis and Applications,1998,132:424-433.
[ 6 ]HU S,Wang H.Convex solutions of boundary value problems arising from Monge-Ampère equations[J].Discrete and Continuous Dynamical Systems,2006,16:705-720.
[ 7 ]WANG F,AN Y.Triple nontrivial radial convex solutions of systems of Monge-Ampère equations[J].Applied Mathematics Letters,2012,25:88-92.
[ 8 ]WANG H.Convex solutions of systems of Monge-Ampère equations[J].Journal of Mathematical Analysis and Applications,2006,318:246-251.
[ 9 ]GUTIERREZ C.The Monge-Ampère Equation[M].Basel:Birkhuser Publishing Ltd,2000.
[10]郭大均.非线性泛函分析[M].3版.济南:山东科学技术出版社,2015.
[11]冯美强,张学梅.Banach空间中非线性常微分方程边值问题[M].北京:科学出版社,2018.