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 $u=(u_1,u_2,\dots .u_n)$, for a class of Kirchhoff-Type Potential Systems with critical exponent, namely $\begin{array}{r}\{\begin{array}{ll}-M_i({\mathcal{A}}_i(u_i))\mathrm{div}({\mathcal{B}}_i(\mathrm{\nabla }{u}_i))=|u_i{|}^{s_i(x)-2}{u}_i+\lambda F_{u_i}(x,u)& \mathrm{in}\mathrm{\Omega },\\ u=0& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where Ω is a bounded smooth domain in ${\mathbb{R}}^N(N\ge 2)$, and ${\mathcal{B}}_i(\mathrm{\nabla }{u}_i)=a_i(|\mathrm{\nabla }{u}_i{|}^{p_i(x)})|\mathrm{\nabla }{u}_i{|}^{p_i(x)-2}\mathrm{\nabla }{u}_i.$ The functions $M_i$, ${\mathcal{A}}_i$, $a_i$ and $a_i$ ($1\le i\le n$) are given functions, whose properties will be introduced hereafter, λ is the positive parameter, and the real function F belongs to $C^1(\mathrm{\Omega }\times {\mathbb{R}}^n)$, $F_{u_i}$ 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 $x\in \overline{\mathrm{\Omega }}$.

COMMUNICATED BY:

• M. A. Ragusa

Keywords:

• Variable exponent spaces
• critical Sobolev exponents
• Kirchhoff-type problems
• p-Laplacian
• $p(x)$-Laplacian
• concentration-compactness principle
• Palais–Smale condition
• Mountain Pass Theorem
• Critical Points Theory

2010 Mathematics Subject Classifications:

• 35B33
• 35D30
• 35J50
• 35J60
• 46E35

Acknowledgments

The author wants to express their gratitude to Professor Abderrahmane EL HACHIMI and to anonymous referee for the careful reading and helpful comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).