  In this paper we define a new operator $J$ for the study of
  $$ \Delta u +f(u)=0,\quad x\in R ^N, N> 2. $$ Using $J$ we can easily see
some qualitative properties of the solutions, for example we can determine how
many times $u$ changes sign, which are the values of the local maxima and
minima, and where $u$ changes concavity.
  We also use this functional to construct nonlinearities $f$ such that this
problem has at least two bound state solutions that change sign $j$ times, for
$j=1,\dots,k-1$. And another $f$ such that this problem has a unique ground
state solution, and at least two bound state solutions that change sign one
time.
