论文标题
具有电势和联合非线性的非线性Schrodinger方程的归一化溶液
Normalized solutions for the nonlinear Schrodinger equation with potential and combined nonlinearities
论文作者
论文摘要
在目前的论文中，我们研究了以下功率非线性的以下非线性schrödinger方程 \ begin {align*} -ΔU+v（x）u+λu= | u |^{2^* - 2} u+μ| u | u |^{q-2} u \ quad \ quad \ quad \ quad \ text {in} \ end {align*} 规定了质量 \ begin {align*} \ int_ {\ mathbb {r}^n} u^2dx = a^2， \ end {align*}其中$μ，a> 0 $，$ q \ in（2，2^*）$，$ 2^*= \ frac {2n} {n-2} $是关键的sobolev endent，$ v $是Infinity的外部电位消失，在Infinity和parame $λ\ in \ Mathbb in \ Mathbb {ragr agr a ragr a r a r}在$ v $的一些轻度假设下，对于$ l^2 $ - 提出的扰动$ q \ in（2，2，2+ \ frac {4} {n} {n}）$，我们证明存在$ a_0> 0 $，以便在上述问题上具有$μ> 0 $ $ $ $ a_ 0 $ a_ 0 $ a_0 $ a_0 $ a_0 $ a_0 $ a_0; a_0 $ a_ 0 \ a_ 0;对于$ l^2 $ - 临界扰动$ q = 2+\ frac {4} {n} $，通过限制$μ$的范围，在pohožaev约束的帮助下，也可以找到上述任何$ a> 0 $的正面状态化解决方案；此外，对于$ l^2 $ -superitality扰动$ q \ in（2+ \ frac {4} {n}，2^*）$，我们通过使用POHOžaev约束来获得上述问题的正面状态化解决方案。 同时，建立了阳性归一化溶液的指数衰减特性，这对于对立波的不稳定性分析很重要。此外，我们对基础状态集进行了描述，并在[2+ \ frac {4} {n} {n}，2^*）$中以$ q \的身份获得了站立波的强不稳定性。本文可以被视为SOAVE的概括[J。功能。肛门。 （2020）]从某种意义上说。
In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities \begin{align*} - Δu+V(x)u+λu=|u|^{2^*-2}u+μ|u|^{q-2}u \quad \quad \text{in} \ \mathbb{ R}^N, \ N\geq 3 \end{align*} having prescribed mass \begin{align*} \int_{ \mathbb{ R}^N}u^2dx=a^2, \end{align*} where $μ, a>0$, $q\in(2, 2^*)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent, $V$ is an external potential vanishing at infinity, and the parameter $λ\in \mathbb{R}$ appears as a Lagrange multiplier. Under some mild assumptions on $V$, for the $L^2$-subcritical perturbation $q\in(2, 2+\frac{4}{N})$, we prove that there exists $a_0>0$ such that the normalized solution with negative energy to the above problem with $μ>0$ can be obtained for $a\in (0, a_0)$; for the $L^2$-critical perturbation $q=2+\frac{4}{N}$, by limiting the range of $μ$, the positive ground state normalized solution to the above problem for any $a>0$ is also found with the aid of the Pohožaev constraint; moreover, for the $L^2$-supercritical perturbation $q\in( 2+\frac{4}{N}, 2^*)$, we get a positive ground state normalized solution for the above problem with $a>0$ and $μ>0$ by using the Pohožaev constraint. At the same time, the exponential decay property of the positive normalized solution is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for $q\in[2+\frac{4}{N}, 2^*)$. This paper can be regarded as a generalization of Soave [J. Funct. Anal. (2020)] in a sense.
