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

Abstract

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 , for a class of Kirchhoff-Type Potential Systems with critical exponent, namely where Ω is a bounded smooth domain in , and The functions , , and () are given functions, whose properties will be introduced hereafter, λ is the positive parameter, and the real function F belongs to , denotes the partial derivative of F with respect to . 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 are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for for all .