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 subject to functional boundary conditions. We obtain a localization of the corresponding non-negative eigenfunctions in terms of their norm. Under additional growth conditions, we also prove the existence of an unbounded set of eigenfunctions for these systems. The class of equations that we study is fairly general and our approach covers some systems of nonlocal elliptic differential equations subject to nonlocal boundary conditions. An example is presented to illustrate the theory.
Highlights
IntroductionA well known result in nonlinear analysis is the Birkhoff-Kellogg invariant-direction
A well known result in nonlinear analysis is the Birkhoff-Kellogg invariant-directionTheorem [1]
In the case of an infinite-dimensional normed linear space V this theorem reads as follows
Summary
A well known result in nonlinear analysis is the Birkhoff-Kellogg invariant-direction. The class of systems occurring in (1) is fairly general and allows us to deal with nonlocal problems of Kirchhoff-type. The formulation of the functionals occurring in (1) allows us to consider multi-point or integral BCs. There exists a wide literature on this topic, for brevity we refer the reader to the recent paper [23]. For further reading on the topics of non-standard elliptic systems and gradient terms appearing within the nonlinearities, we refer the reader to the recent papers [24,25]. Under fairly general conditions, the existence of positive eigenvalues with corresponding non-negative eigenfunctions for the system (1) and illustrate how these results can be applied in the case of nonlocal elliptic systems, see Remark 2. The results complement the ones in [26], by considering more general nonlocal elliptic systems
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