Instability of Standing Waves for INLS with Inverse Square Potential

This work studies an inhomogeneous generalized Hartree equation with inverse square potential. The purpose is to prove the existence and strong instability of inter-critical standing waves. This means that there are infinitely many data near to the ground state, such that the associated solution blows-up in finite time. The proof combines a variational analysis with the standard variance identity. The challenge is to deal with three difficulties: the singular potential |x|−2, an inhomogeneous term |x|−λ, and a non-local source term.