Normalized solutions to Schrödinger equations in the strongly sublinear regime

Abstract

AbstractWe look for solutions to the Schrödinger equation $$\begin{aligned} -\Delta u + \lambda u = g(u) \quad \text {in } \mathbb {R}^N \end{aligned}$$ - Δ u + λ u = g ( u ) in R N coupled with the mass constraint $$\int _{\mathbb {R}^N}|u|^2\,dx = \rho ^2$$ ∫ R N | u | 2 d x = ρ 2 , with $$N\ge 2$$ N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., $$\lim _{s\rightarrow 0}g(s)/s = -\infty $$ lim s → 0 g ( s ) / s = - ∞ , which includes the case $$\begin{aligned} g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \end{aligned}$$ g ( s ) = α s ln s 2 + μ | s | p - 2 s with $$\alpha > 0$$ α > 0 and $$\mu \in \mathbb {R}$$ μ ∈ R , $$2 < p \le 2^*$$ 2 < p ≤ 2 ∗ properly chosen. We consider a family of approximating problems that can be set in $$H^1(\mathbb {R}^N)$$ 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 $$H^1(\mathbb {R}^N)$$ H 1 ( R N ) , we prove the existence of infinitely, many solutions.