Schwarz Symmetrization and Comparison Results for Nonlinear Elliptic Equations and Eigenvalue Problems

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 $L^{\mathrm{\infty }}$ estimates for solutions of nonlinear and eigenvalue problems in terms of other $L^p$ norms.