Abstract

In this paper we define a new operator J for the study of Δu+f(u)=0,x∈RN,N>2. Using J we can easily see some qualitative properties of the solutions, for example we can determine the values where u changes sign, has a local maxima or minima, or changes concavity. We also use this functional to construct, for any k∈N, nonlinearities f such that this problem has at least two bound state solutions that change sign j times, for j=0,1,…,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.