The sharp existence of constrained minimizers for the L²-critical Schrödinger-Poisson system and Schrödinger equations

2017/05/03 by Ye, Hongyu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.01331

Abstract

In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schrödinger-Poisson equations: -Δu+V(x)u+(|x|^-1*u²)u-|u|^(4)/(3)u=λu,~~x∈\R³ on the L²-spheres \widetildeS(c)=\u∈ H¹(\R³)|~∫_\R³V(x)u²dx0\. If V(x)≡0, then by a different method from Jeanjean and Luo [Z. Angrew. Math. Phys. 64 (2013), 937-954], we show that there is no minimizer for all c>0; If 0≤ V(x)∈ L^∞_loc(\R³) and lim_|x|→+∞V(x)=+∞, then a minimizer exists if and only if 0μ₁ for some μ₁>0, then a minimizer exists for each c∈(0,c^*).

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