Positive solutions of nonpositone sublinear elliptic problems

Abstract

Consider the problem $-\Delta u=\lambda f(\cdot, u) $ in $\Omega$, $u=0$ on $\partial\Omega$, $u\gt 0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary when $n\geq2$, $\lambda\gt 0$, and where $f\in C (\overline{\Omega}\times[0,\infty)) $ satisfies $\lim_{s\rightarrow\infty}s^{-p}f(\cdot, s) =\gamma$ for some $p\in(0,1)$ and some $\gamma\in C(\overline{\Omega}) $ such that $\gamma\neq 0$ a.e. in $\Omega$ and, for some positive constants $c$ and $c^{\prime}$, $\gamma^{-}\leq cd_{\Omega}^{\beta}$ for some $\beta\in (\frac{n-1}{n},\infty)$ and $(-\Delta)^{-1}\gamma\geq c^{\prime}d_{\Omega}$, where $d_{\Omega}(x):=dist ( x,\partial \Omega) $ and $\gamma^{-}:=-\min(0,\gamma)$. Under these assumptions we show that for $\lambda$ large enough, the above problem has a positive weak solution $u\in C^{1}(\overline{\Omega})$ such that, for some constant $c^{\prime\prime}\gt 0$, $u\geq c^{\prime\prime}d_{\Omega}$ in $\Omega$.