Multiplicity results for some nonlinear Schrödinger equations with potentials

(K1) K ∈ C(R), K is bounded and K(x) > 0 ∀ x ∈ R. One seeks solutions uε of (NLS) that concentrate, as ε→ 0, near some point x0 ∈ R (semiclassical standing waves). By this we mean that for all x ∈ R \ {x0} one has that uε(x) → 0 as ε→ 0. When K equals a positive constant, say K(x) ≡ 1, (NLS) has been widely investigated, see [2, 3, 10, 12, 15, 16, 18] and references therein. Moreover, the existence of multibump solutions has also been studied in [7, 14], see also [3] where solutions with infinitely many bumps has been proved. It has been also pointed out, see e.g. [2, Section 6], that the results contained in the forementioned papers can be extended to equations where u is substituted by a function g(u), which behaves like u. Nonlinearities depending upon x have been handled in [13, 19] where the existence of one-bump solutions is proved. In a group of papers Equation (NLS) is studied by perturbation arguments. For example, in [2] a Liapunov-Schmidt type procedure is used to reduce, for ε small, (NLS) to a finite dimensional