标题： 具有势的波导流形上的双调和薛定谔方程的归一化解和稳定性 显示英文标题

标题： Normalized solutions and stability for biharmonic Schrödinger equation with potential on waveguide manifold

摘要： 在本文中，我们研究带有势和混合非线性的以下双调和薛定谔方程 \begin{equation*} \left\{\begin{array}{ll}\Delta^2 u +V(x,y)u+\lambda u =\mu|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in \Omega_r \times \mathbb{T}^n, \\ \int_{\Omega_r\times\mathbb{T}^n}u^2dxdy=\Theta,\end{array} \right. \end{equation*} 其中 $\Omega_r \subset \mathbb{R}^d$ 是一个开有界凸区域， $r>0$ 很大且 $\mu\in\mathbb{R}$。 指数满足 $2<p<2+\frac{8}{d+n}<q<4^*=\frac{2(d+n)}{d+n-4}$，使得非线性是质量次临界项和质量超临界项的组合。 在关于 $V(x,y)$ 和 $\mu$的一些假设下，我们得到波导流形上的几个存在性结果。 此外，我们还考虑了解的轨道稳定性。 显示英文摘要

摘要： In this paper, we study the following biharmonic Schr\"odinger equation with potential and mixed nonlinearities \begin{equation*} \left\{\begin{array}{ll}\Delta^2 u +V(x,y)u+\lambda u =\mu|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in \Omega_r \times \mathbb{T}^n, \\ \int_{\Omega_r\times\mathbb{T}^n}u^2dxdy=\Theta,\end{array} \right. \end{equation*} where $\Omega_r \subset \mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $\mu\in\mathbb{R}$. The exponents satisfy $2<p<2+\frac{8}{d+n}<q<4^*=\frac{2(d+n)}{d+n-4}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Under some assumptions on $V(x,y)$ and $\mu$, we obtain the several existence results on waveguide manifold. Moreover, we also consider the orbital stability of the solution.