On semipositone problems over $\mathbb{R}^N$ for the fractional $p$-Laplace operator

Abstract

For $N \geq 1, s\in (0,1)$, 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: \begin{equation}\tag{SP} (-\Delta)_{p}^{s}u = g(x)f_a(u) \text{ in } \mathbb{R}^N, \end{equation} 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 $f$ is a non-negative continuous function on $[0,\infty)$ 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 (SP) 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 $r \in \left[\frac{Np}{N-sp}, \infty\right]$. 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 $g(x) \leq \frac{B}{|x|^{\beta(p-1)+sp}}$ for $B>0, x \neq 0$, and $ \beta \in \left(\frac{N-sp}{p-1}, \frac{N}{p-1}\right)$, we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in $\mathbb{R}^N$.