Entire bounded solutions for a class of sublinear elliptic equations (Q1877867)

The author deals with the question of the existence of a bounded solution to the sublinear elliptic problem \[ \begin{cases} \Delta u=\xi(x) u^\gamma\text{ in }\mathbb{R}^N\\ u\geq 0,\;u\not\equiv 0,\end{cases}\tag{1} \] where \(\xi\in L^\infty_{\text{loc}} (\mathbb{R}^N)\) is nonnegative, \(\gamma \in[0,1]\), and \(\Delta\) denotes as usually the Laplacian on \(\mathbb{R}^N\). By means of thinness at \(\infty\), the author gives a characterization of all functions \(\xi\) for which problem (1) possesses a bounded solution. The author studies (1) for a subdomain \(\Omega\) of \(\mathbb{R}^N\) and establishes necessary and sufficient conditions under which there exists bounded solutions to (1) replacing \(\mathbb{R}^N\) by \(\Omega\).