Quantitative bounds for subcritical semilinear elliptic equations

We prove a priori bounds for weak solutions of semilinear elliptic equations of the form −Δu = cu^p, with 0 < p < p_s = (d + 2)/(d − 2), d ≥ 3, posed on a bounded domain Ω of Rd with boundary conditions u = 0. The bounds are quantitative and we give explicit expressions for all the involved constants. These estimates also allow to compare solutions corresponding to different values of p, an in particular take the limit p → 1. Besides their own interest, these results are useful in the study of the asymptotic convergence with rate of the solutions to the Cauchy-Dirichlet problem for the Fast Diffusion Equation.