We search for non-constant normalized solutions to the semilinear elliptic system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\nu \Delta {v}_{i}+{g}_{i}\left({v}_{j}^{2}\right){v}_{i}={\lambda}_{i}{v}_{i},\phantom{\rule{1em}{0ex}}{v}_{i}>0\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega \hfill \\ {\partial}_{n}{v}_{i}=0\hfill & \text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega \hfill \\ {\int}_{\Omega}{v}_{i}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x=1,\hfill & 1\le i,j\le 2,\phantom{\rule{1em}{0ex}}j\ne i,\hfill \end{array}\right.\end{array}$$ |

$$\begin{array}{c}\hfill {\int}_{\Omega}{v}_{1}{v}_{2}\to 0\phantom{\rule{2em}{0ex}}\text{as}\phantom{\rule{4pt}{0ex}}\nu \to 0.\end{array}$$ |

Accepted:

DOI: 10.1051/cocv/2016028

Keywords: Singularly perturbed problems, normalized solutions to semilinear elliptic systems, multi-population differential games

^{1}; Verzini, Gianmaria

^{2}

@article{COCV_2017__23_3_1145_0, author = {Cirant, Marco and Verzini, Gianmaria}, title = {Bifurcation and segregation in quadratic two-populations mean field games systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1145--1177}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016028}, zbl = {1371.35110}, mrnumber = {3660463}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016028/} }

TY - JOUR AU - Cirant, Marco AU - Verzini, Gianmaria TI - Bifurcation and segregation in quadratic two-populations mean field games systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1145 EP - 1177 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016028/ DO - 10.1051/cocv/2016028 LA - en ID - COCV_2017__23_3_1145_0 ER -

%0 Journal Article %A Cirant, Marco %A Verzini, Gianmaria %T Bifurcation and segregation in quadratic two-populations mean field games systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1145-1177 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016028/ %R 10.1051/cocv/2016028 %G en %F COCV_2017__23_3_1145_0

Cirant, Marco; Verzini, Gianmaria. Bifurcation and segregation in quadratic two-populations mean field games systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1145-1177. doi : 10.1051/cocv/2016028. http://archive.numdam.org/articles/10.1051/cocv/2016028/

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