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

Abstract

espanolContinuamos estudiando la existencia de cotas uniformes a priori para soluciones positivas de equaciones elipticas subcriticas (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 elipticos subcriticos tengan cotas a-priori L∞ en dominios acotados, convexos, y de clase C2. En esta parte II, extendemos nuestros resultados a sistemas elipticos 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 varian sobre la hiperbola critica de Sobolev 1/p+1 + 1/q+1 = N−2/N . Para ecuaciones elipticas 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). Tambien estudiamos el comportamiento asintotico de soluciones radialmente simetric uα = uα(r) de (P)2 cuando α → 0. EnglishWe 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.