Normalized solutions to Schrödinger systems with linear and nonlinear couplings

Abstract

In this paper, we study important Schrödinger systems with linear and nonlinear couplings{−Δu1−λ1u1=μ1|u1|p1−2u1+r1β|u1|r1−2u1|u2|r2+κ(x)u2inRN,−Δu2−λ2u2=μ2|u2|p2−2u2+r2β|u1|r1|u2|r2−2u2+κ(x)u1inRN,u1∈H1(RN),u2∈H1(RN), with the condition∫RNu12=a12,∫RNu22=a22, where N≥2, μ1,μ2,a1,a2>0, β∈R, 2<p1,p2<2⁎, r1,r2>1, r1+r2<2⁎, κ(x)∈L∞(RN) with fixed sign and λ1,λ2 are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for L2-subcritical case when N≥2, and use minimax method to prove this system has a normalized radially symmetric positive solution for L2-supercritical case when N=3, p1=p2=4,r1=r2=2.