Normalized Ground State Solutions for Critical Growth Schrödinger Equations

Abstract

This paper investigates the existence of solutions with a prescribed $L^2$-norm for the nonlinear Sobolev critical Schrödinger equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u =g(u)+|u|^{2^*-2}u, &{}\text {~in~} {\mathbb {R}}^N,\\ \int _{{\mathbb {R}}^{N}}|u|^{2}\textrm{d} x=a, &{} u\in H^1({\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}$$

where $N\ge 3$, $a>0$, $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent, g belongs to the continuous function space ${\mathcal {C}}({\mathbb {R}})$, and the parameter $\lambda$ serving as a Lagrange multiplier. We employ the Sobolev subcritical approximation method to establish the existence of normalized ground state solutions for this particular class of Schrödinger equations with critical growth.