Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent. Issue 4 (4th March 2023) - Chems Eddine, Nabil
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 u = ( u 1, u 2, … . u n ), for a class of Kirchhoff-Type Potential Systems with critical exponent, namely 1 { − 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 2 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 ¯ .