Normalized bound states for the nonlinear Schrodinger equation in bounded domains

2016/07/15 by Pierotti, Dario, Verzini, Gianmaria · 10 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.04520

Abstract

Given ρ>0, we study the elliptic problem find (U,λ)∈ H¹₀(Ω)× ℝ such that \begincases -ΔU+λU=|U|^p-1U ∫_Ω U² dx=ρ, \endcases where Ω⊂ℝ^N is a bounded domain and p>1 is Sobolev-subcritical, searching for conditions (about ρ, N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is L²-subcritical, i.e. 10. In the L²-critical and supercritical case, i.e. when 1+4/N ≤ p < 2^*-1, we show that, for any k∈ℕ, the problem admits solutions having Morse index bounded above by k only if ρ is sufficiently small. Next we provide existence results for certain ranges of ρ, which can be estimated in terms of the Dirichlet eigenvalues of -Δ in H¹₀(Ω), extending to general domains and to changing sign solutions some results obtained in [Noris, Tavares, Verzini, Analysis & PDE, 2014] for positive solutions in the ball.

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