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

Abstract

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: \begin{equation}\tag{SP} (-Δ)_{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 $g(x)\le \frac{B}{|x{|}^{{\hat{I}}^2(p-1)+sp}}$ for $B>0,x\ne 0$, and ${\hat{I}}^2\in \left(\frac{N-sp}{p-1},\frac{N}{p-1}\right)$, we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in ${\mathbb{R}}^N$.

26 pages. This paper expands upon the findings presented in arXiv:2308.00954 from linear to non-linear framework