Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations

The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form $-\mathrm{\Delta }u=c{u}^p$, with $0<p<p_s=(d+2)\mathrm{/}(d-2)$, defined on bounded domains of ${\text{RR}}^d$, $d\ge 3$, without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constant, as well as gradient bounds.