Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights

Marta García-Huidobro, Raúl Manasevich, Satoshi Tanaka

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)293-310
Number of pages18
JournalAdvanced Nonlinear Studies
Volume20
Issue number2
DOIs
Publication statusPublished - 2020 May 1
Externally publishedYes

Keywords

  • A-Priori Bounds
  • Asymptotically Homogeneous
  • Blow-Up,Leray Schauder Degree
  • Quasilinear Elliptic Systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematics(all)

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