The existence of constrained minimizers for a class of nonlinear Kirchhoff–Schrödinger equations with doubly critical exponents in dimension four

In this paper, the existence and nonexistence of energy minimizer of the Kirchhoff–Schrodinger energy function with prescribed L 2 -norm in dimension four are considered. The energy infimum values are completely classified in terms of coefficient and exponent of the nonlinearity. The sharp existence results of global constraint minimizers for both the subcritical and critical exponent cases are obtained, and the criticality is in the sense of both Sobolev embedding and Gagliardo–Nirenberg inequality. Our results also show the delicate difference between the case without a trapping potential function and the one with potential function.