We give two-sided estimates for positive solutions of the superlinearelliptic problem
$-\unicode[STIX]{x1D6E5}u=a(x)|u|^{p-1}u$ with zero Dirichlet boundary condition in a boundedLipschitz domain. Our result improves the well-known a priori
$L^{\infty }$ -estimate and provides information about the boundary decayrate of solutions.