Semilinear elliptic equations

Abstract

This book is based on the lecture notes for a graduate course taught at the Institute of Mathematics of the Federal University of Rio de Janeiro (IM-UFRJ) during the first semester of 2013. Its purpose is to present the classical methods and results for second order monotone nonlinear elliptic equations and for non monotone second order semilinear elliptic equations on a bounded domain, with a special focus on the problem −Δu = λeu. This domain has been extensively studied and there are many books and articles treating different aspects of the problems, some of them are mentioned in the references but this list is from far not exhaustive. The methods and results presented here are not new, but this course gave me the opportunity to put together some known and less known results in a coherent text which is essentially self contained and with all the main proofs. Only some basic results of integration, functional analysis and of spectral decomposition are recalled without proof.