In this paper, the existence of multiple solutions to a nonlinear elliptic equation with a
parameter $\lambda$ is studied. Initially, the existence of two nonnegative solutions is showed for
$0 < \lambda < \hat \lambda$. The first solution has a negative energy while the energy of the
second one is positive for $0 < \lambda < \lambda_0$ and negative for
$\lambda_0 < \lambda < \hat \lambda$.
The values $\lambda_0$ and $\hat \lambda$ are given under
variational form and we show that every
corresponding critical point is solution of the nonlinear
elliptic problem (with a suitable multiplicative term).
Finally, the existence of two classes of infinitely many solutions is showed via the