Abstract
In this paper, we investigate the existence and multiplicity of solutions for a class of fractional $ (p_1(x,.),p_2(x,.)) $-Kirchhoff type problems with Dirichlet boundary data of the following form $ \left( \mathcal{P}_{M_i}^s\right) \; \; \left\{ \begin{array}{lllll} \sum\limits_{i = 1}^{2} M_i \left(\int_{Q}\frac{1}{p_i(x,y)}\frac{|u(x)-u(y)|^{p_i(x,y)}}{|x-y|^{N+sp_i(x,y)}}\; dxdy\right) \left( -\Delta\right)_{p_i(x,.)} ^{s}u(x)\\+\sum\limits_{i = 1}^{2}|u|^{\bar{p}_i(x)-2}u = f(x,u)\quad\rm{in}\; \; \Omega, u = 0 \quad \quad\rm{in}\; \; \; \mathbb{R}^{N}\setminus\Omega. \end{array}\right. $ More precisely, by means of mountain pass theorem with Cerami condition, we show that the above problem has at least one nontrivial solution. Moreover, using Fountain theorem, we prove that $ \left( \mathcal{P}_{M_i}^s\right) $ possesses infinitely many (pairs) of solutions with unbounded energy.
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