*preprint*

**Inserted:** 3 feb 2024

**Last Updated:** 12 jun 2024

**Year:** 2024

**Abstract:**

For $N \ge 1, s\in (0,1)$, and $p \in (1, N/s)$ we find a positive solution to the following class of semipositone problems associated with the fractional $p$-Laplace operator:

$ (-\Delta)_{p}^{s}u = g(x)f_a(u) \text{ in } \mathbb{R}^N, $

where $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is a positive
function, $a>0$ is a parameter and $f_a \in \mathcal{C}(\mathbb{R})$ is defined
as $f_a(t) = f(t)-a$ for $t \ge 0$, $f_a(t) = -a(t+1)$ for $t \in [-1, 0]$, and
$f_a(t) = 0$ for $t \le -1$, where $f \in \mathcal{C}(\mathbb{R}^+)$ satisfies
$f(0)=0$ with subcritical and Ambrosetti-Rabinowitz type growth. Depending on
the range of $a$, we obtain the existence of a mountain pass solution to the problem
in $\mathcal{D}^{s,p}(\mathbb{R}^N)$. Then, we prove mountain pass solutions
are uniformly bounded with respect to $a$, over $L^r(\mathbb{R}^N)$ for every
$r \in [Np/N-sp, \infty]$. In addition, if $p>2N/N+2s$, we establish that the problem
admits a non-negative mountain pass solution for each $a$ near zero. Finally,
under the assumption $g(x) \leq B/

x

^{\beta(p-1)+sp}$ for $B>0, x \neq 0$, and
$ \beta \in (N-sp/p-1, N/p-1)$, we derive an explicit positive subsolution to
the problem and show that the non-negative solution is positive a.e. in
$\mathbb{R}^N$.