A new approach to the study of elliptic semilinear equations

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.