Normalized solutions and stability for biharmonic Schrödinger equation with potential on waveguide manifold
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| Format: | Preprint |
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2024
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| _version_ | 1866909331245498368 |
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| author | Wang, Jun Yin, Zhaoyang |
| author_facet | Wang, Jun Yin, Zhaoyang |
| contents | In this paper, we study the following biharmonic Schrödinger equation with potential and mixed nonlinearities \begin{equation*}
\left\{\begin{array}{ll}Δ^2 u +V(x,y)u+λu =μ|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in Ω_r \times \mathbb{T}^n, \\ \int_{Ω_r\times\mathbb{T}^n}u^2dxdy=Θ,\end{array} \right. \end{equation*} where $Ω_r \subset \mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $μ\in\mathbb{R}$. 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 $μ$, we obtain the several existence results on waveguide manifold. Moreover, we also consider the orbital stability of the solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00032 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions and stability for biharmonic Schrödinger equation with potential on waveguide manifold Wang, Jun Yin, Zhaoyang Analysis of PDEs In this paper, we study the following biharmonic Schrödinger equation with potential and mixed nonlinearities \begin{equation*} \left\{\begin{array}{ll}Δ^2 u +V(x,y)u+λu =μ|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in Ω_r \times \mathbb{T}^n, \\ \int_{Ω_r\times\mathbb{T}^n}u^2dxdy=Θ,\end{array} \right. \end{equation*} where $Ω_r \subset \mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $μ\in\mathbb{R}$. 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 $μ$, we obtain the several existence results on waveguide manifold. Moreover, we also consider the orbital stability of the solution. |
| title | Normalized solutions and stability for biharmonic Schrödinger equation with potential on waveguide manifold |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.00032 |