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 $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0<q<1<p\leq \frac{N+2s}{N-2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $-(-\Delta)^s $, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2}(\frac{N+2s}{N-2s})<q<1$, $N>6s$, $\lambda=1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.