Volume 35, Issue 3
Vector Solutions with Prescribed Component-Wise Nodes for a Schrodinger System

Anal. Theory Appl., 35 (2019), pp. 288-311.

Published online: 2019-04

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• Abstract

For the Schrodinger system

$$\left\{\begin{array}{ll}-\Delta u_j +\lambda_j u_j =\sum_{i=1}^k \beta_{ij} u_i^2 u_j\quad \mbox{in}\ \ \ \ \mathbb R^N,\\ u_j(x)\to0\quad\text{ as }\ \ |x|\to\infty, j=1,\cdots,k,\end{array}\right.$$

where $k\geq 2$ and $N=2, 3$, we prove that for any $\lambda_j>0$ and $\beta_{jj}>0$ and any positive integers $p_j$, $j=1,2,\cdots,k$, there exists $b>0$ such that if $\beta_{ij}=\beta_{ji}\leq b$ for all $i\neq j$ then there exists a radial solution $(u_1,u_2,\cdots,u_k)$ with $u_j$ having exactly $p_j-1$ zeroes. Moreover, there exists a positive constant $C_0$ such that if $\beta_{ij}=\beta_{ji}\leq b\ (i\neq j)$ then any solution obtained satisfies

$$\sum_{i,j=1}^k|\beta_{ij}|\int_{\mathbb R^N}u_i^2u_j^2\leq C_0.$$

Therefore, the solutions exhibit a trend of phase separations as $\beta_{ij}\to-\infty$ for $i\neq j$.

• Keywords

Vector solution, prescribed component-wise nodes, Schrodinger system, variational methods.

35A15, 35J10, 35J50

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@Article{ATA-35-288, author = {}, title = {Vector Solutions with Prescribed Component-Wise Nodes for a Schrodinger System}, journal = {Analysis in Theory and Applications}, year = {2019}, volume = {35}, number = {3}, pages = {288--311}, abstract = {

For the Schrodinger system

$$\left\{\begin{array}{ll}-\Delta u_j +\lambda_j u_j =\sum_{i=1}^k \beta_{ij} u_i^2 u_j\quad \mbox{in}\ \ \ \ \mathbb R^N,\\ u_j(x)\to0\quad\text{ as }\ \ |x|\to\infty, j=1,\cdots,k,\end{array}\right.$$

where $k\geq 2$ and $N=2, 3$, we prove that for any $\lambda_j>0$ and $\beta_{jj}>0$ and any positive integers $p_j$, $j=1,2,\cdots,k$, there exists $b>0$ such that if $\beta_{ij}=\beta_{ji}\leq b$ for all $i\neq j$ then there exists a radial solution $(u_1,u_2,\cdots,u_k)$ with $u_j$ having exactly $p_j-1$ zeroes. Moreover, there exists a positive constant $C_0$ such that if $\beta_{ij}=\beta_{ji}\leq b\ (i\neq j)$ then any solution obtained satisfies

$$\sum_{i,j=1}^k|\beta_{ij}|\int_{\mathbb R^N}u_i^2u_j^2\leq C_0.$$

Therefore, the solutions exhibit a trend of phase separations as $\beta_{ij}\to-\infty$ for $i\neq j$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0009}, url = {http://global-sci.org/intro/article_detail/ata/13117.html} }
TY - JOUR T1 - Vector Solutions with Prescribed Component-Wise Nodes for a Schrodinger System JO - Analysis in Theory and Applications VL - 3 SP - 288 EP - 311 PY - 2019 DA - 2019/04 SN - 35 DO - http://dor.org/10.4208/ata.OA-0009 UR - https://global-sci.org/intro/ata/13117.html KW - Vector solution, prescribed component-wise nodes, Schrodinger system, variational methods. AB -

For the Schrodinger system

$$\left\{\begin{array}{ll}-\Delta u_j +\lambda_j u_j =\sum_{i=1}^k \beta_{ij} u_i^2 u_j\quad \mbox{in}\ \ \ \ \mathbb R^N,\\ u_j(x)\to0\quad\text{ as }\ \ |x|\to\infty, j=1,\cdots,k,\end{array}\right.$$

where $k\geq 2$ and $N=2, 3$, we prove that for any $\lambda_j>0$ and $\beta_{jj}>0$ and any positive integers $p_j$, $j=1,2,\cdots,k$, there exists $b>0$ such that if $\beta_{ij}=\beta_{ji}\leq b$ for all $i\neq j$ then there exists a radial solution $(u_1,u_2,\cdots,u_k)$ with $u_j$ having exactly $p_j-1$ zeroes. Moreover, there exists a positive constant $C_0$ such that if $\beta_{ij}=\beta_{ji}\leq b\ (i\neq j)$ then any solution obtained satisfies

$$\sum_{i,j=1}^k|\beta_{ij}|\int_{\mathbb R^N}u_i^2u_j^2\leq C_0.$$

Therefore, the solutions exhibit a trend of phase separations as $\beta_{ij}\to-\infty$ for $i\neq j$.

Zhaoli Liu & Zhi-Qiang Wang. (2019). Vector Solutions with Prescribed Component-Wise Nodes for a Schrodinger System. Analysis in Theory and Applications. 35 (3). 288-311. doi:10.4208/ata.OA-0009
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