A multiparameter fractional Laplace problem with semipositone nonlinearity

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type\begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u& = & \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&>&0 \text{ in } \Omega\\ u&\equiv &0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document}when the positive parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \mu $\end{document} belong to certain range. Here \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is assumed to be a bounded open set with smooth boundary, \begin{document}$ s\in (0, 1), N> 2s $\end{document} and \begin{document}$ 0<q<1<r\leq \frac{N+2s}{N- 2s}. $\end{document} First we consider \begin{document}$ (P_ \lambda^\mu) $\end{document} when \begin{document}$ \mu = 0 $\end{document} and prove that there exists \begin{document}$ \lambda_0\in(0, \infty) $\end{document} such that for all \begin{document}$ \lambda> \lambda_0 $\end{document} the problem \begin{document}$ (P_ \lambda^0) $\end{document} admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of \begin{document}$ (P_\lambda^0) $\end{document} emanating from infinity. Next for each \begin{document}$ \lambda>\lambda_0 $\end{document} and for all \begin{document}$ 0<\mu<\mu_{\lambda} $\end{document} we establish the existence of at least one positive solution of \begin{document}$ (P_\lambda^\mu) $\end{document} using variational method. Also in the sub critical case, i.e., for \begin{document}$ 1<r<\frac{N+2s}{N-2s} $\end{document} , we show the existence of second positive solution via mountain pass argument.