The Ground State of a Cubic-quintic Nonlinear Schr\"{o}dinger Equation with Radial Potential in the Thomas-Fermi Limit

Abstract

We focus on the ground state of the cubic-quintic nonlinear Schr\"{o}dinger energy functional \begin{gather*} \begin{aligned} {E}(\varphi)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla \varphi|^2+V(x)|\varphi|^2\right)\,dx \pm\frac{1}{4}\int_{\mathbb{R}^d}|\varphi|^4\,dx +\frac{1}{6}\int_{\mathbb{R}^d}|\varphi|^6\,dx, (d=1,2,3) \end{aligned} \end{gather*} under the mass constraint $\int_{\mathbb{R}^d}|\varphi|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $\varphi_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[\mu^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $\mu^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $\varphi_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities.