Eigenvalues of elliptic functional differential systems via a Birkhoff--Kellogg type theorem

Abstract

Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations. An example is presented to illustrate the theory.

Nonzero positive solutions of fractional Laplacian systems with functional terms

“…[17]) to prove our main existence result, namely Theorem 3.3 below; instead, we prove a non-existence result via an elementary argument. In some sense, our existence result stems from a pioneering work by Amman [4,5] and follows a line recently pursued by the authors in the study of elliptic PDEs [9,18,19,20]. We point out that our approach permits to consider (possibly nonlinear) functional BCs: for example, in Section 4 we will discuss the solvability of the following problem:…”