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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 $$\mathrm{i}{\varphi }_t=-▵\varphi +{\left|x\right|}^2\varphi -{\left|x\right|}^b{\left|\varphi \right|}^{p-2}\varphi .$$ 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 $$-\Delta u-\frac{\lambda }{{\left|x\right|}^2}u=u^{2^{*}-1}+\mu u^{-q}\text{in}\Omega \setminus \left\{0\right\}$$ with homogeneous Dirichlet boundary conditions, where $0\in \Omega \subset {ℝ}^N$($N\ge 3$), $0<q<1$, $0<\lambda <{(N-2)}^2/4$ and $2^{*}=2N/(N-2)$. We use variational methods to prove that for suitable $\mu$, the problem has at least two positive weak solutions.