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

• Pardo, Rosa ^[1]
  1. [1] Universidad Complutense de Madrid

     Universidad Complutense de Madrid

     Madrid, España

     
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 37, Nº. 1, 2019 (Ejemplar dedicado a: Revista Integración, temas de matemáticas), págs. 113-148
• Idioma: inglés
• DOI: 10.18273/revint.v37n1-2019006
• Títulos paralelos:
  • Sobre la existencia de cotas a priori para soluciones positivas de problemas elípticos, II
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Continuamos estudiando la existencia de cotas uniformes a priori para soluciones positivas de equaciones elípticas subcríticas  (P)p       − \Delta_pu = f(u), en \Omega,  u = 0,  sobre ∂\Omega, Proporcionamos condiciones suficientes para que las soluciones positivas en C1,μ (\overline{\Omega }) de una clase de problemas elípticos subcríticos tengan cotas a-priori L∞ en dominios acotados, convexos, y de clase C2. En esta parte II, extendemos nuestros resultados a sistemas elípticos Hamiltonianos −\Delta u = f(v), −\Delta v = g(u), en \Omega  , u = v = 0 sobre ∂ \Omega, cuando f(v) = vp/[ln(e + v)]α, g(u) = uq/[ln(e + u)]β, con α, β > 2/(N − 2), y p, q varían sobre la hipérbola crítica de Sobolev 1/p+1 + 1/q+1 = N−2/N . Para ecuaciones elípticas cuasilineales que involucran al operador p-Laplacian, existen cotas a-priori para soluciones positivas de (P)p en el espacio C1,μ(\overline {\Omega }), μ ∈ (0, 1), cuando f(u) = up⋆−1/[ln(e + u)]α, con p∗ = Np/(N − p), y α > p/(N − p). También estudiamos el comportamiento asintótico de soluciones radialmente simétric uα = uα(r) de (P)2 cuando α → 0.

  • English

    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) = vp /[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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