Normalized solutions for nonlinear Schrödinger systems

2015/07/16 by Bartsch, Thomas, Jeanjean, Louis · 3 citations
#35J47 #35P30 #35Q55 #47J30 #58E05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35J50 #Secondary: 35B08

paper · doi:10.48550/arxiv.1507.04649

Abstract

We consider the existence of normalized solutions in H¹(\R^N) × H¹(\R^N) for systems of nonlinear Schrödinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz one is led to coupled systems of elliptic equations of the form \ \beginaligned -\De u₁ amp;= \la₁u₁ + f₁(u₁)+\pa₁F(u₁,u₂),
-\De u₂ amp;= \la₂u₂ + f₂(u₂)+\pa₂F(u₁,u₂),
u₁,u₂amp;∈ H¹(\R^N), N≥2, \endaligned . and we are looking for solutions satisfying ∫_\R^N|u₁|² = a₁, ∫_\R^N|u₂|² = a₂ where a₁>0 and a₂>0 are prescribed. In the system \la₁ and \la₂ are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. f_i(u_i)=μ_i|u_i|^p_i-1u_i, F(u₁,u₂)=\be|u₁|^r₁|u₂|^r₂, with positive constants \be, μ_i, p_i, r_i. The exponents are Sobolev subcritical but may be L²-supercritical: p₁,p₂,r₁+r₂∈]2,2^*[ ∖\2+\frac4N\.

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