In this paper, we establish existence and asymptotic behaviour of nontrivial weak solution of a class of quasilinear stationary Kirchhoff type equations involving the variable exponent spaces with critical growth, namely$$\begin{aligned}{\left\{ \begin{array}{ll} -M (\mathcal{A}(u)) {\rm div} (a(|\nabla u|^{p(x)}) | \nabla u|^{p(x) - 2} \nabla u) = \lambda f (x, u) + |u|^{s(x)-2} u \quad {\rm in} \quad \Omega,\\ u = 0 \quad {\rm on} \quad \partial \Omega,\end{array}\right. } \end{aligned}$$where \({\Omega}\) is a bounded smooth domain of \({\mathbb{R}^N}\) , with homogeneous Dirichlet boundary conditions on \({\partial \Omega}\) , the nonlinearities \({f : \Omega \times \mathbb{R} \rightarrow \mathbb{R}}\) is a continuous function, \({a : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}}\) is a function of the class \({C^1}\) , \({M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}}\) is a continuous function whose properties will be introduced later, and \({\lambda}\) is a positive parameter. We assume that \({\mathcal{C} = \{x \in \Omega : s(x) = \gamma^{*}(x)\} \neq \emptyset}\) , where \({\gamma (x)^{*} = N \gamma (x) / (N - \gamma (x))}\) is the critical Sobolev exponent. We show that the problem has at least one solution, which it converges to zero, in the norm of the space X as \({\lambda \rightarrow + \infty}\) . Our result extends, complement and complete in several ways some of the recent works. We want to emphasize that a difference of some previous research is that the conditions on \({a(\cdot)}\) are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for \({p(x) > 1}\) , for all \({x \in \bar{\Omega}}\) . The main tools used are the Mountain Pass Theorem without the Palais-Smale condition given in [11] and the Concentration Compactness Principle for variable exponent found in [9]. We remark that it will be necessary a suitable truncation argument in the Euler- Lagrange operator associated.

中文翻译：

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具有临界增长指数的Kirchhoff型方程的存在性与渐近性。

在本文中，我们建立了一类涉及具有临界增长变量指数空间的拟线性平稳Kirchhoff型方程的非平凡弱解的存在性和渐近行为，即$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} -M（\ mathcal {A}（u））{\ rm div}（a（| \ nabla u | ^ {p（x）}）| \ nabla u | ^ {p（x）-2} \ nabla u）= \ lambda f（x，u）+ | u | ^ {s（x）-2} u \ quad {\ rm in} \ quad \ Omega，\\ u = 0 \ quad {\ rm on } \ quad \ partial \ Omega，\ end {array} \ right。} \ end {aligned} $$其中\（{\ Omega} \）是\（{\ mathbb {R} ^ N} \）的有界光滑域，在\（{\ partial \ Omega } \），非线性\（{f：\ Omega \ times \ mathbb {R} \ rightarrow \ mathbb {R}} \）是连续函数，\（{a：\ mathbb {R} ^ {+} \ rightarrow \ mathbb {R} ^ {+}} \）是类\（{C ^ 1} \）的函数，\ （{M：\ mathbb {R} ^ {+} _ {0} \ rightarrow \ mathbb {R} ^ {+}} \）是一个连续函数，其属性将在稍后介绍，并且\（{\ lambda} \ ）是一个正参数。我们假设\（{\ mathcal {C} = \ {x \ in \ Omega：s（x）= \ gamma ^ {*}（x）\} \ neq \ emptyset} \），其中\（{\ gamma （x）^ {*} = N \ gamma（x）/（N-\ gamma（x））} \）是关键的Sobolev指数。我们证明问题在空间X的范数中至少有一个解收敛为零，即\（{\ lambda \ rightarrow + \ infty} \）。我们的结果以几种方式扩展，补充和完成了一些近期的工作。我们要强调的是，先前研究的不同之处在于\（{a（\ cdot）} \）的条件足够笼统，可以包含一些令人感兴趣的微分算子。特别地，对于所有\（{x \ in \ bar {\ Omega}} \），我们可以涵盖\（{p（x）> 1} \）的非本地运算符的一般类。所使用的主要工具是[11]中给出的没有Palais-Smale条件的Mountain Pass定理和[9]中发现的可变指数的集中紧凑性原理。我们指出，在关联的Euler-Lagrange运算符中，有必要使用适当的截断参数。