Three solutions for a nonlocal problem with critical growth

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-{\mathrm{\Delta }}_p{)}^{s}u=\mathrm{\mid }u{\mathrm{\mid }}^{p_s^{\ast }-2}u+\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where $(-{\mathrm{\Delta }}_p{)}^s$ is the well known $p$-fractional Laplacian and $p_s^{\ast }=\frac{np}{n-sp}$ is the critical Sobolev exponent for the non local case. The proof is based in the extension of the Concentration Compactness Principle for the $p$-fractional Laplacian and Ekeland’s variational Principle.