By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential $$ {\rm i}\varphi _t=-\triangle \varphi +|x|^2\varphi -|x|^b|\varphi |^{p-2}\varphi . $$ We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.
Consider a class of elliptic equation of the form −∆u − λ ⁄ |x| 2 u = u 2 ∗−1 + µu −q in Ω \ {0} with homogeneous Dirichlet boundary conditions, where 0 ∈ Ω ⊂ RN (N ≥ 3), 0 < q < 1, 0 < λ < (N − 2)2 /4 and 2∗ = 2N/(N − 2). We use variational methods to prove that for suitable µ, the problem has at least two positive weak solutions.