A multiparameter semipositone fractional laplacian problem involving critical exponent

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type $$(P_{\lambda }^{\mu })\{\begin{array}{lll}(-\mathrm{\Delta }{)}^{s}u& =& \lambda (u^q-1)+\mu u^{r}in\mathrm{\Omega }\\ u& >& 0in\mathrm{\Omega }\\ u& \equiv & 0on{\mathbb{R}}^N\setminus \mathrm{\Omega }\mathrm{.}\end{array}$$ when the positive parameters $\lambda$ and $\mu$ belongs to certain range. Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is assumed to be a bounded open set with smooth boundary, $s\in (0,1),N>2s$ and $0<q<1<r\le \frac{N+2s}{N-2s}\mathrm{.}$ The proof relies on the construction of a positive subsolution for $(P_{\lambda }^0)$ for $\lambda >{\lambda }_0\mathrm{.}$ Now for each $\lambda >{\lambda }_0,$ for all small $0<\mu <{\mu }_{\lambda }$ we establish the existence of at least one positive solution of $(P_{\lambda }^{\mu })$ using variational method. Also in the subcritical case, i.e., for $1<r<\frac{N+2s}{N-2s}$, we show the existence of second positive solution via mountain pass argument.