On the normalized eigenvalue problems for nonlinear elliptic operators (Q868765)

Let \(D\) be a bounded open in a real Banach space \(X\). This paper is devoted to the study of the nonlinear eigenvalue problem \(Au-C(\lambda ,u)=0\) in \(D\), \(u=0\) on \(\partial D\), where \(\lambda\geq 0\), \(A\) is an unbounded maximal monotone operator on \(X\), and \(C\) is defined and continuous on \(\mathbb R_+\times\partial D\) such that zero is not in the weak closure of a subset of \(\{C(\lambda ,u)/\| C(\lambda ,u)\| \}\). The main result of the present paper establishes that, under some natural assumptions, this nonlinear eigenvalue problem does not depend on any properties of \(C\) located in \(\mathbb R_+\times D\). The main idea of the proof consists in the construction of a continuous extension of \(C\) based on its values on \(\overline{\mathbb R_+}\times\partial D\) such that its new range inherits the weak nonconvexity property. An application to nonlinear elliptic equations under degenerate conditions is provided in the last section of the paper.