Positive Solutions for Fractional p- Laplace Semipositone Problem with Superlinear Growth

Abstract

We consider a semipositone problem involving the fractional $p$ Laplace operator of the form \begin{equation*} \begin{aligned} (-Δ)_p^s u &=μ( u^{r}-1) \text{ in } Ω,\\ u &>0 \text{ in }Ω,\\ u &=0 \text{ on }Ω^{c}, \end{aligned} \end{equation*} where $\Omega$ is a smooth bounded convex domain in ${\mathbb{R}}^N$, $p-1<r<p_s^{\ast }-1$, where $p_s^{\ast }:=\frac{Np}{N-ps}$, and $\mu$ is a positive parameter. We study the behaviour of the barrier function under the fractional $p$-Laplacian and use this information to prove the existence of a positive solution for small $\mu$ using degree theory. Additionally, the paper explores the existence of a ground state positive solution for a multiparameter semipositone problem with critical growth using variational arguments.

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