The asymptotic behaviors of normalized ground states for nonlinear Schrödinger equations

Abstract

In this paper, we study the relation between the least energy levels and between the minimizers of the following minimization problems

$$\begin{aligned} E_\sigma (\rho )=\inf \Big \{\frac{1}{2}\int _{\mathbb {R}^N} |\nabla w|^2 -\frac{1}{2\sigma +2}\int _{\mathbb {R}^N}|w|^{2\sigma +2} \Big | \int _{\mathbb {R}^N} w^2=\rho \Big \} \end{aligned}$$

and

$$\begin{aligned} Z(\rho ) =\inf \Big \{\frac{1}{2}\int _{\mathbb {R}^N} |\nabla w|^2 -\frac{1}{2}\int _{\mathbb {R}^N}w^{2}\log w^2 \Big | \int _{\mathbb {R}^N} w^2=\rho \Big \}. \end{aligned}$$

We show that as $\sigma \rightarrow 0^+$, the minimizers for $E_\sigma (\rho )$, after rescaling, converge to the minimizers of $Z(\rho )$. Besides, we also give estimates for $E_\sigma (\rho )$ and the corresponding Lagrange multiplier when $\sigma$ is small.