Normalized solutions to a non-variational Schr ̈odinger system

Prof. Andrzej Szulkin
Stockholm University

Abstract: The elliptic system of 2 equations

−∆u_i+κ_iu_i=μ_iu_i³+2λu_iu_j², i,j=1,2,i̸=j,

u_i ∈ H₁(R^N), |u_i|₂ =r_i >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

−∆u_i+κ_iu_i=μ_iu_i^p+λ_iju_j^α_iju_j^β_ij, i,j=1,2,i̸=j,

u_i >0, u_i ∈ H₁(R^N), |u_i|²=r_i>0, i=1,2.

Here 2≤N≤4, 1+4/N<p<(N+2)/(N −2)₊, μ_i>0, λ_ij>0 for all i,j or λ_ij<0 for all i, j, α_ij≥1, β_ij>0 and α_ij+β_ij<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.