Normalized solutions and stability for biharmonic Schr\"odinger equation with potential on waveguide manifold

Abstract

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.