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

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

Resumen

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.

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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Publicado
2019-02-19