On semipositone problems over RN for the fractional p-Laplace operator

For $N\ge 1,s\in (0,1)$, and $p\in (1,\frac{N}{s})$ we find a positive solution to the following class of semipositone problems associated with the fractional p-Laplace operator:(SP)${(-\Delta )}_p^{s}u=g\left(x\right)f_a\left(u\right)\text{ in }{R}^N,$ where $g\in L^1\left(R^N\right)\cap L^{\infty }\left(R^N\right)$ is a positive function, $a>0$ is a parameter and $f_a\in C\left(R\right)$ is defined as $f_a\left(t\right)=f\left(t\right)-a$ for $t\ge 0$, $f_a\left(t\right)=-a(t+1)$ for $t\in [-1,0]$, and $f_a\left(t\right)=0$ for $t\le -1$, where f is a non-negative continuous function on $[0,\infty )$ satisfies $f\left(0\right)=0$ with subcritical and Ambrosetti-Rabinowitz type growth. Depending on the range of a, we obtain the existence of a mountain pass solution to (SP) in $D^{s,p}\left(R^N\right)$. Then, we prove mountain pass solutions are uniformly bounded with respect to a, over $L^r\left(R^N\right)$ for every $r\in [\frac{Np}{N-sp},\infty ]$. In addition, if $p>\frac{2N}{N+2s}$, we establish that (SP) admits a non-negative mountain pass solution for each a near zero. Finally, under the assumption $g\left(x\right)\le \frac{B}{|x{|}^{\beta (p-1)+sp}}$ for $B>0,x\ne 0$, and $\beta \in (\frac{N-sp}{p-1},\frac{N}{p-1})$, we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in $R^N$.