On a “zero mass” nonlinear Schrödinger equation

We look for positive solutions to the nonlinear Schr¨odinger equation −\varepsilon^2\Delta u − V (x)f'(u) = 0 in RN, where V is a continuous bounded positive potential and f satisfies particular growth conditions which make our problem fall in the so called “zero mass case”. We prove an existence result for any \varepsilon > 0, and a multiplicity result for \varepsilon sufficiently small