Entire solutions of sublinear elliptic equations in anisotropic media

Abstract

We study the nonlinear elliptic problem $-\Delta u=\rho (x)f(u)$ in $\RR^N$ ($N\ge 3$), $\lim\_{|x|\ri\infty}u(x)=\ell$, where $\ell \ge 0$ is a real number, $\rho (x)$ is a nonnegative potential belonging to a certain Kato class, and $f(u)$ has a sublinear growth. We distinguish the cases $\ell >0$ and $\ell =0$ and we prove existence and uniqueness results if the potential $\rho (x)$ decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.