Normalized grounded states for a coupled nonlinear schr\"odinger system on R³

We investigate the existence of normalized ground states to the system of coupled Schr\"odinger equations: equationeq: 0. 1 cases - u₁ + ₁ u₁ = ₁ |u₁|^p₁-2u₁ + r₁|u₁|^r₁-2u₁|u₂|^r₂ & in R^3, - u₂ + ₂ u₂ = ₂|u₂|^p₂-2u₂ + r₂|u₁|^r₁|u₂|^r₂-2u₂ & in R³, cases equation subject to the constraints S₀䃑 S₀䃒 = \ (u₁ H¹ (R³) ) |ₑ℃ u₁² dx = a₁²\ \ (u₂ H¹ (R³) ) |ₑ℃ u₂² dx = a₂²\, where ₁, ₂ > 0, r₁, r₂ > 1, and 0. Our focus is on the coupled mass super-critical case, specifically, 103. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable.