We propose a simple minimization method to show the existence of least energy solutions to the normalized problem$\{\begin{array}{c}-\mathrm{\Delta }u+\lambda u=g(u)\mathrm{in}{\mathbb{R}}^N,N\ge 3,\hfill \\ u\in H^1({\mathbb{R}}^N),\hfill \\ {\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=\rho >0,\hfill \end{array}$ where ρ is prescribed and $(\lambda ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball in $L^2({\mathbb{R}}^N)$ of radius ρ is demonstrated, which allows to provide general growth assumptions imposed on g. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at 0.

我们提出了一种简单的最小化方法，以显示归一化问题的最小能量解的存在$\{\begin{array}{c}-\mathrm{\Delta }ü+\lambda ü=G（ü）\mathrm{在}{\mathbb{[R}}^{ñ}，ñ\ge 3，\hfill \\ ü\in H^{\mathrm{1个}}（{\mathbb{[R}}^{ñ}），\hfill \\ {\int }_{{\mathbb{[R}}^{ñ}}|ü{|}^{\mathrm{2个}}dX=\rho >0，\hfill \end{array}$其中ρ是规定的，并且$（\lambda ，ü）\in \mathbb{[R}\times H^{\mathrm{1个}}（{\mathbb{[R}}^{ñ}）$待定。基于Nehari和Pohozaev约束线性组合的能量函数直接最小化的新方法与封闭球相交。${\mathrm{大号}}^{\mathrm{2个}}（{\mathbb{[R}}^{ñ}）$证明了ρ的半径，可以提供对g施加的一般增长假设。到目前为止，我们涵盖了文献中考虑的最著名的物理示例和非线性，并且我们承认质量临界增长为0。