Normalized solutions to Schrödinger equations in the strongly sublinear regime.

We look for solutions to the Schrödinger equation - Δ u + λ u = g (u) in R N coupled with the mass constraint ∫ R N | u | 2 d x = ρ 2 , with N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., lim s → 0 g (s) / s = - ∞ , which includes the case g (s) = α s ln s 2 + μ | s | p - 2 s with α > 0 and μ ∈ R , 2 < p ≤ 2 ∗ properly chosen. We consider a family of approximating problems that can be set in H 1 (R N) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of H 1 (R N) , we prove the existence of infinitely, many solutions. [ABSTRACT FROM AUTHOR]