Existence and Asymptotic Behaviour for a Kirchhoff Type Equation With Variable Critical Growth Exponent

Abstract

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.