Normalized ground state solutions for a class of nonlinear elliptic systems
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초록
We apply the minimization method to show the existence of the least energy solutions to the normalized problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u-\mu_{1}\Delta v+\lambda_1 v=f(v)~~\mbox{in}~~\mathbb{R}^{N},\\ -\Delta v-\mu_{2}\Delta u+\lambda_2 u=g(u)~~\mbox{in}~~\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}|u|^{2}dx=a_1>0, \displaystyle\int_{\mathbb{R}^{N}}|v|^{2}dx=a_2>0, \end{array} \right. \end{equation*} where $N\geq3$, $\mu_{1}$ and $\mu_{2}$ are nonnegative numbers, and $\mu_{1}\mu_{2}>1$, $a_1>0, a_2>0$ are prescribed, $\lambda_{1},~\lambda_{2}\in\mathbb{R}$ are Lagrange multipliers, and $(\lambda_{1},u,\lambda_{2},v)\in\mathbb{R}\times H^1(\mathbb{R}^N)\times \mathbb{R}\times H^1(\mathbb{R}^N)$. The approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed balls in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ of radius $a_{1}$ and $a_{2}$ are demonstrated, which allows providing general growth assumptions imposed on $f$ and $g$.
Abstract
We apply the minimization method to show the existence of the least energy solutions to the normalized problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u-\mu_{1}\Delta v+\lambda_1 v=f(v)~~\mbox{in}~~\mathbb{R}^{N},\\ -\Delta v-\mu_{2}\Delta u+\lambda_2 u=g(u)~~\mbox{in}~~\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}|u|^{2}dx=a_1>0, \displaystyle\int_{\mathbb{R}^{N}}|v|^{2}dx=a_2>0, \end{array} \right. \end{equation*} where $N\geq3$, $\mu_{1}$ and $\mu_{2}$ are nonnegative numbers, and $\mu_{1}\mu_{2}>1$, $a_1>0, a_2>0$ are prescribed, $\lambda_{1},~\lambda_{2}\in\mathbb{R}$ are Lagrange multipliers, and $(\lambda_{1},u,\lambda_{2},v)\in\mathbb{R}\times H^1(\mathbb{R}^N)\times \mathbb{R}\times H^1(\mathbb{R}^N)$. The approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed balls in $L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ of radius $a_{1}$ and $a_{2}$ are demonstrated, which allows providing general growth assumptions imposed on $f$ and $g$.