Study of fractional semipositone problems on $\mathbb{R}^N$

Abstract

Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-\Delta)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.