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

• 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. 77-111
• Idioma: inglés
• DOI: 10.18273/revint.v37n1-2019005
• Títulos paralelos:
  • Sobre la existencia de cotas a priori para soluciones positivas de problemas elípticos, I
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Este artículo proporciona un estudio sobre la existencia de cotas a priori uniformes para soluciones positivas de problemas elípticos subcríticos (P)p        -\Delta_p u =f(u),  en  \Omega,    u = 0, sobre \partial\Omega ampliando el rango conocido de no-linealudades subcríticas para las que las soluciones positivas están acotadas a priori. Nuestros argumentos se apoyan en el método de ‘moving planes’, la identidad de Pohozaev, resultados de regularidad en W1,q para q > N, y el Teorema de Morrey. En esta parte I, cuando p = 2 demostramos que existen cotas a priori para soluciones positivas clásicas de (P)2 con f(u) = u2∗−1/[ln(e+u)]α, siendo 2∗ = 2N/(N−2), y para α> 2/(N − 2). Consideramos también dominios no-convexos, recurriendo a la transformada de Kelvin.

    En un siguiente artículo, parte II, extendemos nuestros resultados para sistemas elípticos Hamiltonianos (ver [22]) y al p-Laplacian (ver [10]). También estudiamos el comportamiento asintótico de las soluciones radialmente simétricas uα = uα(r) de (P)2 cuando α → 0 (ver [24]).

  • English

    This paper gives a survey over the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations(P)p     -\Delta_p u =f(u),  in  \Omega,    u = 0, on \partial\Omegawidening the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded. Our arguments rely on the moving planes method, a Pohozaev identity, W1,q regularity for q > N, and Morrey’s Theorem. In this part I, when p = 2, we show that there exists a-priori bounds for classical, positive solutions of (P)2 with f(u) = u2∗−1/[ln(e + u)]α, with 2∗ = 2N/(N − 2), and α > 2/(N − 2). Appealing to the Kelvin transform, we cover non-convex domains.

    In a forthcoming paper containing part II, we extend our results for Hamiltonian elliptic systems (see [22]), and for the p-Laplacian (see [10]). We also study the asymptotic behavior of radially symmetric solutions uα = uα(r) of (P)2 as α → 0 (see [24]).

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