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 $$ (P_\lambda^\mu)\left\{ \begin{array}{lll} (-\Delta)^s u&=& \lambda(u^{q}-1)+\mu u^r \mbox{ in } \Omega\\ u&>&0 \mbox{ in } \Omega\\ u&\equiv &0 \mbox{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $$ when the positive parameters $\lambda$ and $\mu$ belongs to certain range. Here $\Omega\subset\mathbb R^N$ is assumed to be a bounded open set with smooth boundary, $s\in (0,1), N> 2s$ and $0<q<1<r\leq \frac{N+2s}{N- 2s}.$ The proof relies on the construction of a positive subsolution for $(P_\lambda^0)$ for $\lambda>\lambda_0.$ 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.