In this paper, we establish existence of infinitely many weak solutions
for a class of quasilinear stationary Kirchhoff type equations, which
involves a general variable exponent elliptic operator with critical
growth. Precisely, we study the following nonlocal problem
\begin{equation*} \begin{cases}
-\displaystyle{M}(\mathscr{A}(u))\operatorname{div}\Bigl(a(|\nabla
u|^{p(x)})|\nabla
u|^{p(x)-2}\nabla u\Bigl) =
\lambda f(x,u)+ |u|^{s(x)-2}u
\text{ in }\Omega,
\\ u = 0 \text{ on }
\partial \Omega,
\end{cases} \end{equation*} where
$\Omega$ is a bounded smooth domain of
$\mathbb{R}^N,$ with homogeneous Dirichlet
boundary conditions on $\partial
\Omega,$ the nonlinearity
$f:\overline{\Omega}\times
\mathbb{R}\to
\mathbb{R}$ is a continuous function,
$a:\mathbb{R}^{+}\to\mathbb{R}^{+}$
is a function of the class $C^{1},$
$M:\mathbb{R}^{+}_{0}\to\mathbb{R}^{+}$
is a continuous function, whose properties will be introduced later,
$\lambda$ is a positive parameter and
$p,s\in
C(\overline{\Omega})$. We assume that
$\mathscr{C}=\{x\in
\Omega:
s(x)=\gamma^{*}(x)\}\neq
\emptyset,$ where
$\gamma^{*}(x)=N\gamma(x)/(N-\gamma(x))$
is the critical Sobolev exponent. We will prove that the problem has
infinitely many solutions and also we obtain the asymptotic behavior of
the solution as $\lambda\to 0^{+}$.
Furthermore, we emphasize that a difference with previous researches is
that the conditions on $a(\cdot)$ are general overall
enough to incorporate some interesting differential operators. Our work
covers a feature of the Kirchhoff’s problems, that is, the fact that the
Kirchhoff’s function $M$ in zero is different from zero, it also
covers a wide class of nonlocal problems for $p(x)>1,$
for all $x\in
\overline{\Omega}.$ The main tool to
find critical points of the Euler Lagrange functional associated with
this problem is through a suitable truncation argument,
concentration-compactness principle for variable exponent found in