Normalized solutions of nonlinear Schr\"odinger equations

TL;DR

This paper investigates the existence of infinitely many solutions to a class of nonlinear Schrödinger equations with superlinear, subcritical nonlinearities, especially in cases where the associated energy functional is unbounded below.

Contribution

It establishes the existence of infinitely many solutions for nonlinear Schrödinger equations with nonhomogeneous nonlinearities, even when the energy functional is not bounded below.

Findings

Proved existence of infinitely many solutions.

Handled cases with nonhomogeneous, odd nonlinearities.

Addressed unbounded below energy functionals.

Abstract

We consider the problem -\Delta u - g(u) = \lambda u, u \in H^1(\R^N), \int_{\R^N} u^2 = 1, \lambda\in\R, in dimension $N\ge2$. Here $g$ is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the $L^2$-unit sphere, and we show the existence of infinitely many solutions.