Sobre la existencia de cotas a priori para soluciones positivas de problemas elípticos, I

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

Resumen

Este artículo proporciona un estudio sobre la existencia de cotas a priori uniformes para soluciones positivas de problemas elípticos subcríticos

(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]).

Palabras clave: Estimaciones a priori, no-linealidades subcríticas, método de ‘moving planes’, igualdad de Pohozaev, hipérbola crítica de Sobolev, bifurcación biparamétrica

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Citas

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Publicado
2019-02-19