Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

Abstract

The orbital stability of standing waves of nonlinear Schrodinger equations with a general nonlinear term is investigated in this paper. We study the corresponding minimizing problem with L2-constraint: $$E_\alpha = \inf\left\{\frac{1}{2}\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx - \int\limits_{\mathbb{R}^N} F(|u|) dx; u \in H^1(\mathbb{R}^N), \|u\|_{L^2(\mathbb{R}^N)}^2=\alpha\right\}.$$ We discuss when a minimizing sequence with respect to Eα is precompact. We prove that there exists α0 ≥ 0 such that there exists a global minimizer if α > α0 and there exists no global minimizer if α 0 are established, and the existence results with respect to \({E_{\alpha_0}}\) under some conditions are obtained.