We consider the following nonlinear Schrodinger equation in $${\mathbb {R}}^N(N\ge 2)$$ : $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=g(u), \\ u\in H^1({\mathbb {R}}^N), ~\mathop {\int }\nolimits _{{\mathbb {R}}^N} u^2=c, \end{array}\right. \end{aligned}$$ where $$c>0$$ is a given constant, $$\lambda \in {\mathbb {R}}$$ is a Lagrange multiplier, and $$g\in C^1({\mathbb {R}},{\mathbb {R}})$$ . We deal with the case where the associated functional is not bounded from below on the $$L^2$$ sphere $$S(c)=\left\{ u \in H^{1}\left( {\mathbb {R}}^{N}\right) :\mathop {\int }\nolimits _{{\mathbb {R}}^N} u^2=c\right\} $$ . We show that the ground state energy is strictly decreasing with respect to c. Then we apply this property to give a new proof for the existence of ground state solutions via minimizing methods. We also obtain some other properties of the ground state energy.

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薛定谔方程归一化解的新观察

我们在 $${\mathbb {R}}^N(N\ge 2)$$ 中考虑以下非线性薛定谔方程： $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda u=g(u), \\ u\in H^1({\mathbb {R}}^N), ~\mathop {\int }\nolimits _{{\mathbb {R}}^N } u^2=c, \end{array}\right。\end{aligned}$$ 其中 $$c>0$$ 是一个给定的常数，$$\lambda \in {\mathbb {R}}$$ 是一个拉格朗日乘数，而 $$g\in C^1( {\mathbb {R}},{\mathbb {R}})$$ 。我们处理关联泛函在 $$L^2$$ 球体 $$S(c)=\left\{ u \in H^{1}\left( {\mathbb { R}}^{N}\right) :\mathop {\int }\nolimits _{{\mathbb {R}}^N} u^2=c\right\} $$ 。我们表明基态能量相对于 c 严格减少。然后我们应用这个性质通过最小化方法给出基态解存在的新证明。