Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

Abstract

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -Δu + λu = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = ρ> 0, \end{cases} where $\rho$ is prescribed and $(\lambda ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$.

to appear in Journal of Functional Analysis

References in corpus (2)

• A mass supercritical problem revisited
• On the symmetry of minimizers

Cited by in corpus (5)

• A mass supercritical problem revisited
• Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation
• Normalized ground state solutions of nonlinear Schrödinger equations involving exponential critical growth
• On the Kirchhoff equation with prescribed mass and general nonlinearities
• Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities