Existence of strictly positive solutions for sublinear elliptic problems in bounded domains

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb{R}}^N$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[0,\infty \right)\to \left[0,\infty \right)$ be a continuous function such that $k_1{\xi }^p\le f\left(\xi \right)\le k_2{\xi }^p$ for all $\xi \ge 0$ and some $k_1,k_2>0$ and $p\in \left(0,1\right)$. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right)f\left(u\right)$ in $\Omega$, $u=0$ on $\partial \Omega$.

To appear in Advanced Nonlinear Studies