The Ground State of a Cubic-quintic Nonlinear Schrödinger Equation with Radial Potential in the Thomas-Fermi Limit

We focus on the ground state of the cubic-quintic nonlinear Schrödinger 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 ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }\phi {\mathrm{\mid }}^{2}dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C\mathrm{\mid }x{\mathrm{\mid }}^p(p\ge 2)$ as $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, including the harmonic potential. When $N\to +\mathrm{\infty }$, we show that up to a suitable scaling the ground state ${\phi }_N$ 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 $u^{TF}(x)={[{\mu }^{TF}-C_0\mathrm{\mid }x{\mathrm{\mid }}^p]}_{+}^{\frac{1}{4}}$, 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 $\epsilon$ 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 $N\to +\mathrm{\infty }$. Finally, we establish a new energy method to obtain the $L^{\mathrm{\infty }}$-convergence rates of ground states ${\phi }_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities.