Workshop on Pdes 2022 - Szulkin

Normalized solutions to a non-variational Schr ̈odinger system

Prof. Andrzej Szulkin
Stockholm University

Date: 16th February 10:00 am - 10:45 am (UTC -3) Link: PPGM/UFSCar Youtube channel

Abstract: The elliptic system of 2 equations

−∆uiiuiiui3+2λuiuj2, i,j=1,2,i̸=j,

ui ∈ H1(RN), |ui|2 =ri >0, i=1,2

has been recently studied in dimensions N ≤ 4. Here ri are prescribed and κi are free (they appear as Lagrange multipliers). Such systems arise e.g. when studying mixtures of Bose-Einstein condensates or propagation of wave packets in nonlinear optics. The L2-norms respectively represent the number of particles and power supply. Also various extensions (nonlinearities other than cubic, l instead of 2 equations) have been recently studied. In all results we know of the system is variational.

In this talk we will be concerned with a system of l equations which in general is non-variational. If l=2, the system we consider is

−∆uiiuiiuipijujαijujβij, i,j=1,2,i̸=j,

ui >0, ui ∈ H1(RN), |ui|2=ri>0, i=1,2.

Here 2≤N≤4, 1+4/N<p<(N+2)/(N −2)+, μi>0, λij>0 for all i,j or λij<0 for all i, j, αij≥1, βij>0 and αijij<p, or in some cases, ≤ p. We show, using a combination of variational and topological arguments, that this system possesses at least one solution.

This is joint work with Mónica Clapp.