标题： $\mathbb{R}^3$中势函数的 Kirchhoff 型方程的归一化解 显示英文标题

标题： Normalized solutions for a Kirchhoff type equations with potential in $\mathbb{R}^3$

摘要： 在本文中，我们研究如下Kirchhoff型方程\begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*}在归一化约束条件$\displaystyle\int_{\R^3}u^2=c$下的归一化解的存在性，其中$a,b,c>0$是给定的常数，而非线性项$g(u)$是非常一般的且质量超临界。 在对$V(x)$和$g(u)$作出一些适当的假设下，我们可以证明对于任何给定的$c>0$，存在基态归一化解$(u_c, \lambda_c)\in H^1(\R^3)\times\mathbb{R}$。由于非局部项的存在，任何$(PS)_C$序列$\{w_n\}$的弱极限$u$可能不属于相应的 Pohozaev 流形，这与局部问题不同。 因此，我们必须克服一些新的困难以获得$(PS)_C$序列的紧性。 显示英文摘要

摘要： In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying the normalized constraint $\displaystyle\int_{\R^3}u^2=c$, where $a,b,c>0$ are prescribed constants, and the nonlinearities $g(u)$ are very general and of mass super-critical. Under some suitable assumptions on $V(x)$ and $g(u)$, we can prove the existence of ground state normalized solutions $(u_c, \lambda_c)\in H^1(\R^3)\times\mathbb{R}$, for any given $c>0$. Due to the presence of the nonlocal term, the weak limit $u$ of any $(PS)_C$ sequence $\{w_n\}$ may not belong to the corresponding Pohozaev manifold, which is different from the local problem. So we have to overcome some new difficulties to gain the compactness of a $(PS)_C$ sequence.