On the normalized eigenvalue problems for nonlinear elliptic operators
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.
scientific article; zbMATH DE number 5129632