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

TL;DR

This paper investigates the existence and stability of normalized solutions to a biharmonic Schrödinger equation with potential and mixed nonlinearities on waveguide manifolds, revealing conditions for solutions and their stability.

Contribution

It introduces new existence and stability results for biharmonic Schrödinger equations with mixed nonlinearities on waveguides, considering potential effects and large domain parameters.

Findings

Existence of solutions under certain potential and parameter conditions.

Orbital stability of solutions established.

Solutions exist for a range of nonlinear exponents.

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.