Nonzero positive solutions of nonlocal elliptic systems with functional BCs

Abstract

We discuss the existence and non-existence of non-negative weak solutions for second order nonlocal elliptic systems subject to functional boundary conditions. Our approach is based on classical fixed point index theory combined with some recent results by the author.2010 Mathematics Subject Classification. Primary 35J47, secondary 35B09, 35J57, 35J60, 47H10.

Eigenvalues of Elliptic Functional Differential Systems via a Birkhoff–Kellogg Type Theorem

“…Here we discuss, 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. Our results are new and complement previous results of the author [23], by allowing the presence of gradient terms within the nonlinearities and the functionals. The results also complement the ones in [26], by considering more general nonlocal elliptic systems.…”

“…The framework of (1) allows us to deal with non-homogenous BCs of functional type. In the case of nonlocal elliptic equations, non-homogeneous BCs have been investigated by Wang and An [21], Morbach and Corrẽa [22] and by the author [23]. The formulation of the functionals occurring in (1) allows us to consider multi-point or integral BCs.…”

“…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] and references therein. 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].…”

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

“…When dealing with systems of second-order BVPs, the functional terms w i occurring in (4) can be used to incorporate the nonlocalities that appear in the differential equations, while the functionals h ij originate directly from the BCs. In the context of positive solutions, the idea of incorporating the nonlocal terms of differential equations within the nonlinearities has been exploited in the case of equations by Fijałkowski and Przeradzki [4] and Enguiça and Sanchez [5], while the case of systems of second-order elliptic operators has been considered by the author [6,7]. We seek solutions of the system (4) in a product of cones of a kind that differs from (3); in particular, we work on products of cones in the space C 1 [0, 1] where the functions are positive on a subinterval of [0, 1] and are allowed to change sign elsewhere, this follows the line of research initiated by the author and Webb in [8].…”

Asymptotic properties of PDEs in compact spaces

“…Even then, there has been a recent effort to overcome this difficulties, mainly by imposing some kind of symmetry on the operator that defines the equation and, in particular, searching for radial solutions [7,8,[17][18][19][20][21]28]. More general approaches also appear for elliptic PDEs and systems of PDEs [24][25][26].…”