Recent results on quasilinear differential equations. II. (Q2846677)

This paper concerns the Fredholm alternative for the \(p\)-Laplacian at the first eigenvalue. It is mainly a survey paper of joint results of the author published in [\textit{P. Drábek} et al., Indiana Univ. Math. J. 53, No. 2, 433--482 (2004; Zbl 1081.35031)]. It is very interesting to note that the author can avoid technicalities concerning sets where the gradient of the principal eigenfunction of the \(p\)-Laplacian is zero using a more recent result [\textit{H. Lou}, Chin. Ann. Math., Ser. B 29, No. 5, 521--530 (2008; Zbl 1153.35044)]. The main tool to prove existence and multiplicity results close to the principal eigenvalue is ``the bifurcation from infinity'' argument. In order to study local behavior (in the \(\lambda \)-direction) of the bifurcation, asymptotics of large solutions for \(\lambda \) near the principal eigenvalue are derived. It is interesting that the direction of the bifurcation is opposite in the cases \(1<p<2\) and \(p>2\).NEWLINENEWLINE For Part I see [Zbl 1289.35114].NEWLINENEWLINEFor the entire collection see [Zbl 1262.35002].