Normalized solutions to Schrödinger equations in the strongly sublinear regime

Abstract

We look for solutions to the Schrödinger equation \[ -Δu + λu = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint ${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx={\rho }^2$, with $N\ge 2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., ${\lim }_{s\to 0}g(s)/s=-\infty$, which includes the case \[ g(s) = αs \ln s^2 + μ|s|^{p-2} s \] with $\alpha >0$ and $\mu \in \mathbb{R}$, $2<p\le 2^{\ast }$ properly chosen. We consider a family of approximating problems that can be set in $H^1({\mathbb{R}}^N)$ 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 $H^1({\mathbb{R}}^N)$, we prove the existence of infinitely many solutions.

References in corpus (1)

• On the symmetry of minimizers

Cited by in corpus (1)

• Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities