  Results about existence of a signed ground state solution and multiple
solutions (if $f$ is odd with respect to the second variable) are proven for a
class of asymptotically linear elliptic problems involving a Carath\'eodory
type nonlinearity satisfying assumptions weaker than (fF) in \cite{GLZ} or
$(F_{2})_{+}$ in \cite{CM}. A close relation between the behaviour of
$\lim_{|t|\to \infty}f(x, t)/|t|$ and the number of solutions is stablished.
