A dimension-depending multiplicity result for the Schrödinger equation

We consider the Schrödinger equation { −∆u+ V (x)u = λK(x)f(u) in R ; u ∈ H(R ), (Pλ) where N ≥ 2, λ ≥ 0 is a parameter, V,K : R → R are radially symmetric functions, and f : R → R is a continuous function with sublinear growth at infinity. We first prove that for λ small enough no non-zero solution exists for (Pλ), while for λ large enough at least two distinct non-zero radially symmetric solutions do exist. By exploiting a Ricceri-type three-critical points theorem, the principle of symmetric criticality and a group-theoretical approach, the existence of at least N − 3 (N mod 2) distinct pairs of non-zero solutions is guaranteed for (Pλ) whenever λ is large enough, N 6= 3, and f is odd.