Structure of solutions set of nonlinear eigenvalue problems
The authors use degree theory and global bifurcation arguments in order to obtain information concerning the structure of the solution set for some nonlinear eigenvalue problems. This work goes back until \textit{P. H. Rabinowitz}'s global bifurcation theorems giving the alternative between unbounded continua of solutions and closed loops [J. Funct. Anal. 7, 487--513 (1971; Zbl 0212.16504)]. Some general results in both directions are proved here by using degree theory and set theoretic topology theorems. An application is given to the a problem \(-\Delta u=\mu m(x)u+ b(x)u^a\) in \(\Omega\), \(\partial u/\partial n= 0\) on \(\partial\Omega\), where \(m\) and \(b\) change sign on \(\Omega\), \(m\) changes sign on \([x\in\Omega: b(x)>0]\) and also \(1< a<(N+2)/(N-2)\). In particular, the authors prove the existence of loops improving previous work by \textit{K. J. Brown} [J. Differ. Equations 239, No. 2, 296--310 (2007; Zbl 1331.35129)].
scientific article; zbMATH DE number 6522721
scientific article