GROUNDSTATES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL VANISHING AT INFINITY

Abstract. Groundstates of the stationary nonlinear Schrödinger equa-tion −∆u+ V u = Kup−1, are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special care is paid to the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in L2(RN) is studied and decay estimates are derived using Moser iteration scheme. The results depend on whether V decays slower than |x|−2 at infinity. Content