  We compare the distribution function and the maximum of solutions of
nonlinear elliptic equations defined in general domains with solutions of
similar problems defined in a ball using Schwarz symmetrization. As an
application, we prove the existence and bound of solutions for some nonlinear
equation. Moreover, for some nonlinear problems, we show that if the first
$p$-eigenvalue of a domain is big, the supremum of a solution related to this
domain is close to zero. For that we obtain $L^{\infty}$ estimates for
solutions of nonlinear and eigenvalue problems in terms of other $L^p$ norms.
