A multiparameter semipositone fractional laplacian problem involving critical exponent

Abstract

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type $$ (P_λ^μ)\left\{ \begin{array}{lll} (-Δ)^s u&=& λ(u^{q}-1)+μu^r \mbox{ in } Ω\\ u&>&0 \mbox{ in } Ω\\ u&\equiv &0 \mbox{ on }{\mathbb R^N\setminusΩ}. \end{array}\right. $$ 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 $0<q<1<r\le \frac{N+2s}{N-2s}.$ 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.