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We compare the distribution function and the maximum of solutions ofnonlinear elliptic equations defined in general domains with solutions ofsimilar problems defined in a ball using Schwarz symmetrization. As anapplication, we prove the existence and bound of solutions for some nonlinearequation. 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 thisdomain is close to zero. For that we obtain $L^{\infty}$ estimates forsolutions of nonlinear and eigenvalue problems in terms of other $L^p$ norms.

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