Multiple solutions of systems involving fractional Kirchhoff-type equations with critical growth

Abstract

In this paper we are going to study existence and multiplicity of solutions of a system involving fractional Kirchhoff-type and critical growth of form Mathematics subject classification (2010): 35J60, 35J70, 58E05.

Nonzero positive solutions of fractional Laplacian systems with functional terms

“…Among others, let us mention here the equations, driven by the fractional Laplacian, which have additional nonlocal terms and are often referred to as Kirchhoff‐type equations . For instance, Kirchhoff‐type equations on bounded domains have been recently studied in [8, 11, 15, 28], while systems on bounded domains are investigated, for example, in [12]. Moreover, Kirchhoff‐type equations on the whole of $$\mathbb {R}^n$$ have been studied in [3, 6, 7, 25].…”

“…Let us mention here, for instance, the recent papers by Alves, de Lima, and Nóbrega [1, 2]: In these papers, the authors obtained Rabinowitz‐type global bifurcation results of positive solutions of a parametric fractional Laplacian equation in $$\mathbb {R}^n$$ using the Leray–Schauder degree. On the other hand, due to the presence of the nonlocal functional weights, system (1.1) can be viewed as a nonlinear fractional Kirchhoff‐type problem , even if in a slightly different direction than the one proposed in [12, 15].…”

“…12) where 𝐵 1 is the Euclidean unit ball, and osc 𝐵 1 (⋅) is as in (4.2). Clearly, problem (4.12) is of the form (3.1), with 1.…”

Nonzero positive solutions of fractional Laplacian systems with functional terms

“…Among others, let us mention here the equations, driven by the fractional Laplacian, which have additional non-local terms and are often referred to as Kirchhoff-type equations. For instance, Kirchhoff-type equations on bounded domains have been recently studied in [8,11,15,25], while systems on bounded domains are investigated, e.g., in [12]. Moreover, Kirchhoff-type equations on the whole of R n have been studied in [3,6,7,22].…”

“…Let us mention here, for instance, the recent papers by Alves, de Lima and Nóbrega [1,2]: in these papers, the authors obtained Rabinowitz-type global bifurcation results of positive solutions of a parametric fractional Laplacian equation in R n using the Leray-Schauder degree. On the other hand, due to the presence of the non-local functional weights, system (1.1) can be considered as a nonlinear fractional Kirchhoff-type problem, even if in a slightly different direction than the one proposed in [12,15].…”