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Annales Henri Poincaré

, Volume 19, Issue 10, pp 3031–3051 | Cite as

Elliptic Systems with Some Superlinear Assumption Only Around the Origin

  • Patricio Cerda
  • Marco Aurelio Souto
  • Pedro Ubilla
Article
  • 30 Downloads

Abstract

In this paper, using a priori bound techniques we study existence of positive solutions of the elliptic system:
$$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }(|x|^{\alpha _1}\nabla u) = |x|^{\beta _1} f(|x|,u,v) \ \ x \in B, \\ -\text{ div }(|x|^{\alpha _2}\nabla v) = |x|^{\beta _2} g(|x|,u,v) \ \ x \in B, \\ u(x) = 0 =v(x), \ \ \ x \in \partial B. \end{array} \right. \end{aligned}$$
where B is the unitary ball centered at the origin. Assuming that fg are nonnegative nonlinearities and that \(f(|x|,u,v)+g(|x|,u,v)\) is superlinear at 0 and at \(\infty \), we establish some results of existence of one positive solution. As an application, we establish two positive solutions for some non-homogeneous elliptic system. The main novelties here are that the nonlinearities could have growth above the critical hyperbola on some part of the domain as well as only local superlinear hypotheses at \(\infty \)

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Patricio Cerda
    • 1
  • Marco Aurelio Souto
    • 2
  • Pedro Ubilla
    • 1
  1. 1.Departamento de Matemática y C. C.Universidad de Santiago de ChileSantiagoChile
  2. 2.Departamento de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil

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