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.