Multiple solutions for a critical Kirchhoff system

Existence of positive solutions for a class of Kirchhoff type systems involving critical exponents

“…for all (s, t) ∈ R 2 and all x ∈ Ω, where θ i ∈ (0, 1) and C is a positive constant. From (18) and (M 0 ), for all w = (u, v) ∈ H, we get…”

“…It should be noticed that we don't need any conditions on the Kirchhoff function M(t) except for the boundedness from below on [0, +∞) as stated in (M 0 ). So, our situation introduced here is different from those presented in [3,17,18]. We also refer the interested readers to some results [23,28,31] in which the authors considered the problem in the special case M(t) = a + bt, a > 0 and b ≥ 0.…”

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

“…(3) The case off i not depending on Du, withw i and h i acting on the cone of non-negative functions C(Ω, R n + ), has been studied by the author in [23]. The following example provides a system of the type (11) that cannot be handled by the theory of [14][15][16][17][18][19], due to the presence of gradient terms in the nonlinearities, and by the results in [20], due to the presence of the nonlocal BCs. It also illustrates, in contrast to previous results on Kirchhofftype systems known to the author, that it is possible to consider some interaction between the gradient terms of the components of the system occurring within the nonlocal part of the differential equation or within the nonlocal BCs.…”

“…The approach employed in [14] is the sub-supersolution method. A similar approach has also been used in the recent papers [15,16], while variational methods have been utilized in [17][18][19].…”

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

“…The approach employed in [14] is the sub-supersolution method. The system (1.3) has been studied also by the sub-supersolution method in [8,35,6,7], while variational methods were employed in [19,32,38,49].…”