Abstract
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 ∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L p norms.
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Bonorino, L.P., Montenegro, J.F.B. Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems. Annali di Matematica 192, 987–1024 (2013). https://doi.org/10.1007/s10231-012-0255-0
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DOI: https://doi.org/10.1007/s10231-012-0255-0
Keywords
- Schwarz symmetrization
- Distribution function
- Nonlinear elliptic problem
- Eigenvalue problem
- Optimal estimates
- Degenerate elliptic equation