Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} $\mathcal{L}$_{K} u + \mu, |u|^{q-1}u + \lambda,|u|^{p-1}u &=& 0 \quad$\text{in}$\quad \Omega, \[2mm] u&=&0 \quad\mbox{in}\quad$\mathbb{R}$^N\setminus\Omega, \end{eqnarray*} where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N>2s$, $s\in (0,1)$, $0<q<1<p\le \frac{N+2s}{N-2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $-(-\mathrm{\Delta }{)}^s$, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2}(\frac{N+2s}{N-2s})<q<1$, $N>6s$, $\lambda =1$, we find ${\mu }^{\ast }>0$ such that for any $\mu \in (0,{\mu }^{\ast })$, there exists at least one sign changing solution.