extremality conditions
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2019 ◽  
Vol 19 (4) ◽  
pp. 693-715 ◽  
Author(s):  
Haijun Luo ◽  
Zhitao Zhang

AbstractWe study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling {\beta>0}, we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants {\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as {\beta\to\infty}, we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.


2019 ◽  
Vol 83 (2) ◽  
pp. 253-271
Author(s):  
Erkka Haapasalo ◽  
Juha-Pekka Pellonpää

Analysis ◽  
2017 ◽  
Vol 37 (3) ◽  
Author(s):  
Rajendra Dahal ◽  
Stanley Drvol ◽  
Christopher S. Goodrich

AbstractWe present some initial results for detecting extrema of a function


2012 ◽  
Vol 53 (12) ◽  
pp. 122203 ◽  
Author(s):  
Anna Jenčová

2008 ◽  
Vol 77 (8) ◽  
Author(s):  
Ivan Booth ◽  
Stephen Fairhurst

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