当前位置: X-MOL 学术Adv. Nonlinear Stud. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights
Advanced Nonlinear Studies ( IF 2.1 ) Pub Date : 2020-05-01 , DOI: 10.1515/ans-2020-2082
Marta García-Huidobro 1 , Raúl Manasevich 2 , Satoshi Tanaka 3
Affiliation  

Abstract In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

中文翻译:

具有非齐次算子和权重的拟线性方程组的正解

摘要 在本文中,我们处理包含强非线性算子的边值问题的正径向对称解。我们给出的正解存在的证明使用爆炸法作为搜索解的先验界限的主要成分。膨胀论证是一个接一个的矛盾,并使用了一种缩放,让人想起最小曲面理论中使用的缩放,参见 [B. Gidas 和 J. Spruck,非线性椭圆方程正解的先验界限,通讯。偏微分方程 6 1981, 883–901],因此算子、拉普拉斯算子或 p-拉普拉斯算子以及二元幂或幂函数的齐次性在该方法中发挥着基本作用。因此,当微分算子不再齐次时,对于第二个成员也是如此,应用爆破法来获得解的先验界限似乎是一项几乎不可能完成的任务。尽管如此,在[M. García-Huidobro、I. Guerra 和 R. Manásevich,通过爆炸弱耦合系统的正径向解的存在,Abstr。应用程序 肛门。3 1998, 1-2, 105-131],我们能够克服这个困难并获得特定(更简单)类型问题的先验界限。我们在本文中展示了渐近齐次函数在同样的意义上提供了非线性重新缩放,这使我们能够将爆破方法推广到我们目前的情况。在获得先验界限后,解的存在性遵循 Leray-Schauder 拓扑度理论。García-Huidobro、I. Guerra 和 R. Manásevich,通过爆炸弱耦合系统的正径向解的存在,Abstr。应用程序 肛门。3 1998, 1-2, 105-131],我们能够克服这个困难并获得特定(更简单)类型问题的先验界限。我们在本文中展示了渐近齐次函数在同样的意义上提供了非线性重新缩放,这使我们能够将爆破方法推广到我们目前的情况。在获得先验界限后,解的存在性遵循 Leray-Schauder 拓扑度理论。García-Huidobro、I. Guerra 和 R. Manásevich,通过爆炸弱耦合系统的正径向解的存在,Abstr。应用程序 肛门。3 1998, 1-2, 105-131],我们能够克服这个困难并获得特定(更简单)类型问题的先验界限。我们在本文中展示了渐近齐次函数在同样的意义上提供了非线性重新缩放,这使我们能够将爆破方法推广到我们目前的情况。在获得先验界限后,解的存在性遵循 Leray-Schauder 拓扑度理论。我们在本文中展示了渐近齐次函数在同样的意义上提供了非线性重新缩放,这使我们能够将爆破方法推广到我们目前的情况。在获得先验界限后,解的存在性遵循 Leray-Schauder 拓扑度理论。我们在本文中展示了渐近齐次函数在同样的意义上提供了非线性重新缩放,这使我们能够将爆破方法推广到我们目前的情况。在获得先验界限后,解的存在性遵循 Leray-Schauder 拓扑度理论。
更新日期:2020-05-01
down
wechat
bug