L 2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

This article studies the Schrödinger equation with an inhomogeneous combined term i ∂ t u − ( − Δ ) s u + λ 1 | x | − b | u | p u + λ 2 | u | q u = 0 , where s ∈ ( 1 2 , 1 ) , λ 1 , λ 2 = ± 1 , 0 < b < { 2 s , N } and p , q > 0 . We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p , q , λ 1 and λ 2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L 2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.