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A NOTE ON NORMALISED GROUND STATES FOR THE TWO-DIMENSIONAL CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 09 October 2023, pp. 552-561
- Print publication:
- June 2024
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- Article
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We consider the two-dimensional minimisation problem for
$\inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$, where the energy functional 
$ E_a(\varphi )$ is a cubic-quintic Schrödinger functional defined by 
$E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi |^6\,dx$. We study the existence and asymptotic behaviour of the ground state. The ground state 
$\varphi _{a}$ exists if and only if the 
$L^2$ mass a satisfies 
$a>a_*={\lVert Q\rVert }^2_2$, where Q is the unique positive radial solution of 
$-\Delta u+ u-u^3=0$ in 
$\mathbb {R}^2$. We show the optimal vanishing rate 
$\int _{\mathbb {R}^2}|\nabla \varphi _{a}|^2\,dx\sim (a-a_*)$ as 
$a\searrow a_*$ and obtain the limit profile.
