Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$; $N>1$ with a smooth boundary or $\Omega=(0,1)$. We study positive solutions to the boundary value problem of the form: \begin{equation*} \begin{aligned} -\Delta_p u - \Delta_q u&=\lambda f(u) &&\mbox{in}~\Omega,\\ u &= 0 &&\mbox{on}~\partial\Omega, \end{aligned} \end{equation*} where $q \in [2,p)$, $\lambda$ is a positive parameter, and $f:[0,\infty) \mapsto \mathbb{R}$ is a class of $C^1$, non-decreasing and $p$-sublinear functions at infinity (i.e. $\lim_{t \rightarrow \infty} \frac{f(t)}{t^{p-1}}=0$) that are negative at the origin (semipositone). We discuss the existence of positive solutions for $\lambda\gg1$. Further, when $p=4,q=2$, $\Omega=(0,1)$ and $f(s)=(s+1)^\gamma-2$; $\gamma \in (0,3)$, we provide the exact bifurcation diagram for positive solutions. In particular, we observe two positive solutions for a finite range of $\lambda$ and a unique positive solution for $\lambda\gg1.$

Keywords

• $p$–$q$ laplacian
• semipositone problems
• positive solutions