Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave–convex nonlinearities

Abstract

We consider the combined effect of concave-convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When α + β < 4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais-Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.

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].…”