In any dimension $N\ge 1$, for given mass $m>0$ and when the $C^1$ energy functional $$I(u):=\frac{1}{2}{\int }_{{ℝ}^N}|\nabla u{|}^{2}dx-{\int }_{{ℝ}^N}F(u)dx$$ is coercive on the mass constraint $$S_m:=\left\{\right.u\in H^1({ℝ}^N)|\parallel u{\parallel }_{L^2({ℝ}^N)}^2=m\left\}\right.,$$ we are interested in searching for constrained critical points at positive energy levels. Under general conditions on $F\in C^1(ℝ,ℝ)$ and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic–quintic nonlinear Schrödinger equation in ${ℝ}^3$.

中文翻译：

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矫顽问题的正能量归一化解及其在三次-五次非线性薛定谔方程中的应用

在任何维度$ñ\ge 1$, 对于给定的质量$米>0$当$C^1$能量泛函$$我(你)：=\frac{1}{2}{\int }_{{ℝ}^{ñ}}|\nabla 你{|}^{2}dX-{\int }_{{ℝ}^{ñ}}F(你)dX$$对质量约束是强制性的$${\mathrm{小号}}_{米}：=\left\{\right.你\in H^1({ℝ}^{ñ})|\parallel 你{\parallel }_{{\mathrm{大号}}^2({ℝ}^{ñ})}^2=米\left\}\right.,$$我们有兴趣在正能量水平上寻找受约束的临界点。在一般条件下$F\in C^1(ℝ,ℝ)$对于合适的质量范围，我们设法构建了这样的临界点，这些临界点表现为局部最小化或对应于山口或对称的山口水平。特别是，我们的结果揭示了三次 - 五次非线性薛定谔方程${ℝ}^3$.