Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents

In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrodinger equations involving the mass and energy critical exponents. To show the existence of a ground state, we solve a minimization problem related to the virial identity, so that we need to compare the minimization value to the best constant of the Gagliardo–Nirenberg inequality because our nonlinearities contain the mass critical nonlinearity. Once we obtain the ground state, we can introduce a subset $${\mathcal {A}}_{\omega , -}$$ of $$H^{1}({\mathbb {R}}^d)$$ for each $$\omega > 0$$ as in Berestycki and Cazenave (C R Acad Sci Paris Ser I Math 293:489–492, 1981). Then, it turn out that any radial solution starting from $${\mathcal {A}}_{\omega , -}$$ blows up in a finite time.