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(del u(i))) = vertical bar u(i)vertical bar(si(x)-2) u(i) + lambda F-ui (x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N(N >= 2), and B-i(del u(i)) = a(i)(vertical bar del u(i)vertical bar(pi(x)))vertical bar del u(i)vertical bar(pi(x)-2)del 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, lambda is the positive parameter, and the real function F belongs to C-1(Omega x R-n), F-ui 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 is an element of (Omega) over bar.