In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrödinger 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 Sér 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.

在本文中，我们研究了涉及质量和能量临界指数的一类非线性Schrödinger方程的基态和爆燃问题的存在。为了显示基态的存在，我们解决了与病毒身份有关的最小化问题，因此我们需要将最小化值与Gagliardo-Nirenberg不等式的最佳常数进行比较，因为我们的非线性包含质量临界非线性。一旦我们得到基态时，我们可以引入一个子集\（{\ mathcal {A}} _ {\欧米加， - } \）的\（H ^ {1}（{\ mathbb {R}} ^ d）\ ），如Berestycki和Cazenave中所述（\ omega> 0 \）（CR Acad Sci ParisSérI Math 293：489–492，1981）。然后，发现从\（{\ mathcal {A}} _ {\ omega，-} \）在有限时间内爆炸。