Positive solutions to sublinear elliptic problem

Abstract

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega$ in ${\mathbb{R}}^d$, $d\ge 3$, such that $L1\le 0$. We study existence and properties of continuous solutions to the following problem \begin{equation}\label{00} Lu=φ(\cdot,u),% & \hbox{in $\Omega$; in the sens of distribution;} \\ \end{equation} in $\Omega ,$ where $\Omega$ is a Greenian domain for $L$ {(possibly unbounded)} in ${\mathbb{R}}^d$ and $\phi$ is a nonnegative function on $\Omega \times [0,+\infty [$ increasing with respect to the second variable. By means of thinness, we obtain a characterization of $\phi$ for which \eqref{00} has a nonnegative nontrivial bounded solution.

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• Sublinear elliptic problems under radiality. Harmonic $NA$ groups and Euclidean spaces