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Vector solutions with prescribed component-wise nodes for a Schrodinger system


Academic Report

Title: Vector solutions with prescribed component-wise nodes for a Schrodinger system

Reporter: LIU Zhaoli (Capital Normal University)

Time: October 12, 2018 (Friday) AM 8:30-9:30

Location: A1101# room, Innovation Park Building

Contact: DAI Guowei (tel:84708351-8135)


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

The brief introduction to the reporter: Professor Liu's main research directions are nonlinear functional analysis. He has made many outstanding achievements in variational methods and elliptic partial differential equations. He worked as a Humboldt scholar for two years at Giessen University in Germany. He was supported by the National Science Foundation for Outstanding Youth in 2008. In 2009, he was appointed Professor Jiang Scholar, Minister of Education. He has published more than 70 SCI papers in top international journals such as "Adv. Math.", "Comm. Math. Phys", "Indiana Univ. Math. J.", "Comm. Partial Differential Equations", "J. Funct. Anal.", "Proc. London Math. Soc.", "Calc. Var. Partial Differential Equations", "Ann. Inst. H. Poincaré Anal. Non Linéaire"Math. Z. and J. Differential Equations", and these papers have been cited more than 1200 times by domestic and foreign counterparts.