Revista Integración, temas de matemáticas.
Vol. 37 No. 1 (2019): Revista Integración, temas de matemáticas
Research and Innovation Articles

On the existence of a priori bounds for positive solutions of elliptic problems, II

Rosa Pardo
Universidad Complutense de Madrid, Departamento de Análisis Matemático y Matemática Aplicada, Madrid, Spain.

Published 2019-02-19

Keywords

  • A priori estimates,
  • subcritical nonlinearity,
  • moving planes method,
  • Pohozaev identity,
  • critical Sobolev hyperbola,
  • biparameter bifurcation
  • ...More
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How to Cite

Pardo, R. (2019). On the existence of a priori bounds for positive solutions of elliptic problems, II. Revista Integración, Temas De matemáticas, 37(1), 113–148. https://doi.org/10.18273/revint.v37n1-2019006

Abstract

We continue studying the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations
(P)p       − \Delta_pu = f(u), in \Omega,  u = 0, on ∂\Omega,
We provide sufficient conditions for having a-priori L bounds for C1,μ (\overline{\Omega }) positive solutions to a class of subcritical elliptic problems in bounded, convex, C2 domains. In this part II, we extend our results to Hamiltonian elliptic systems −\Delta u = f(v),−\Deltav = g(u), in \Omega, u = v = 0 on ∂\Omega, when f(v) = v/[ln(e + v)]α, g(u) = uq/[ln(e + u)]β, with α, β > 2/(N − 2), and p, q are lying in the critical Sobolev hyperbolae 1/p+1 + 1/q+1 = N−2/N . For quasilinear elliptic equations involving the p-Laplacian, there exists a-priori bounds for positive solutions of (P)p when f(u) = up⋆−1/[ln(e + u)]α, with p = Np/(N−p), and α > p/(N−p). We also study the asymptotic behavior of radially symmetric solutions uα = uα(r) of (P)2 as α → 0.

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