Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent.

In this paper, by using the concentration-compactness principle of Lions for variable exponents found in [Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. Electron J Differ Equ. 2010;141:1–18.] and the Mountain Pass Theorem without the Palais–Smale condition given in [Rabinowitz PH. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.], we obtain the existence and multiplicity solutions u = (u 1 , u 2 , .... u n) , for a class of Kirchhoff-Type Potential Systems with critical exponent, namely { − M i ( A i (u i)) div ( B i (∇ u i)) = | u i | s i (x) − 2 u i + λ F u i (x , u) in Ω , u = 0 on ∂ Ω , where Ω is a bounded smooth domain in R N (N ≥ 2) , and B i (∇ u i) = a i ( | ∇ u i | p i (x) ) | ∇ u i | p i (x) − 2 ∇ u i. The functions M i , A i , a i and a i ( 1 ≤ i ≤ n) are given functions, whose properties will be introduced hereafter, λ is the positive parameter, and the real function F belongs to C 1 (Ω × R n) , F u i denotes the partial derivative of F with respect to u i . Our results extend, complement and complete in several ways some of many works in particular [Chems Eddine N. Existence of solutions for a critical (p1(x),... , pn(x))-Kirchhoff-type potential systems. Appl Anal. 2020.]. We want to emphasize that a difference of some previous research is that the conditions on a i (.) are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for p i (x) > 1 for all x ∈ Ω ¯ . [ABSTRACT FROM AUTHOR]