Nonzero positive solutions of a multi-parameter elliptic system with functional BCs

Abstract

Abstract. We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multipoint, integral and nonlinear boundary conditions. We also present a non-existence result.We provide some examples to illustrate the applicability our theoretical results.

Nonzero positive solutions of nonlocal elliptic systems with functional BCs

“…In what follows the closure and the boundary of subsets of a coneP are understood to be relative toP . With these ingredients we can now state a result regarding the existence of positive solutions for the system (1.4), that extends the results of Theorem 2.4 of [23] to this new setting. In the sequel we denote by1 the constant function equal to 1 onΩ.…”

“…Our results are new and complement the results of [30], by considering non-radial cases, by allowing the presence of functional BCs and by permitting, in the non-local terms of differential equations, an interaction between all the components of the system. We also improve the results in [23] in the case of local elliptic equations, by weakening the assumptions on the BCs.…”

Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs

“…here, in (1.3), αijk , βijkl are non-negative coefficients and ω i , τ i ∈ O while, in (1.4), αjk , βjkl are non-negative continuous functions on O. In particular we observe that nonlinear, nonlocal BCs have seen recently attention in the framework of elliptic equations: we refer the reader to the papers [5,6,14,15,17,18,24] and references therein. We wish to point out that an advantage of our setting, with respect to the theory developed in [5,6,14,15,17,24], is the possibility to allow also gradient dependence within the functionals occurring in the BCs.…”

Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

“…f , g : [0, 1] × (-∞, +∞) × (-∞, +∞) → (-∞, +∞) are continuous functions, k : [0, +∞) → [0, +∞) is a continuous function, and D α 0 + is the Riemann-Liouville fractional derivative of order α. Fractional differential equations have been increasingly adopted to describe some physical phenomena in thermology, electromagnetic wave, electrochemistry, and other applications [1][2][3][4][5][6][7][8]. There are a great deal of results about the existence and uniqueness of positive solutions for fractional boundary value problems.…”