AI 中文总结
本文研究带短程势的非紧度量图上L²-超临界非线性薛定谔方程，结合山路几何等方法证明了归一化解的存在性，是该领域的首次相关结果。
AI 中文摘要
我们关注非紧度量图G上L²-超临界非线性薛定谔方程归一化解的存在性，该方程满足质量约束∫_G |u|²dx=μ>0，其中λ为拉格朗日乘子，方程形式为：在G的每条边e上，-u''+W(x)u+λu=χ(x)|u|^{p-2}u；在每个顶点v∈V处，满足∑_{e∋v}u'_e(v)=0。此处p>6，χ是紧核𝒦的特征函数，非线性项具有局域性，势函数W有界、非负，且在每条无界边上趋于无穷时消失，至少有一条无界边满足∫_{1}^{∞}xW(x)dx<∞。对于任意μ>0，我们得到一个λ>0的正解，它是严格正能量水平下的约束临界点。我们的方法结合了一族逼近泛函的一致山路几何、带莫尔斯指数类型信息的单调性技巧，以及排除拉格朗日乘子发散的爆破分析。关键在于乘子的严格正性，为此我们构造了一个平方反比衰减的势，其在半直线上存在零能L²解。据我们所知，这是首次在存在外势的非紧度量图上得到L²-超临界非线性薛定谔方程归一化解的存在性结果。
英文摘要
We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schrödinger equation on a noncompact metric graph $G$, \[ \begin{cases} -u''+W(x)u+λu=χ(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases} \] under the mass constraint $\int_G |u|^2\,dx=μ>0$, where $λ$ arises as a Lagrange multiplier. Here $p>6$, $χ$ is the characteristic function of the compact core $\mathcal K$, so that the nonlinearity is localized, and the potential $W$ is bounded, nonnegative and vanishing at infinity along every unbounded edge, with $\int_{1}^{\infty}xW(x)\, dx<\infty$ on at least one of them. For every $μ>0$ we obtain a positive solution with $λ>0$, arising as a constrained critical point at a strictly positive energy level. Our approach combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick with Morse index type information, and a blow-up analysis ruling out the divergence of the Lagrange multipliers. A key point is the strict positivity of the multiplier, for which we exhibit a potential of inverse-square decay admitting a zero-energy $L^2$ solution on a half-line. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.
