Multiple Normalized Solutions for Nonlinear Biharmonic Schrödinger Equations in ℝN with L2-Subcritical Growth

Pub Date : 2024-06-01 DOI:10.1007/s10255-024-1131-6
Jun Wang, Li Wang, Ji-xiu Wang
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Abstract

In this article, we consider the existence of normalized solutions for the following nonlinear biharmonic Schrödinger equations

$$\left\{{\matrix{{{\Delta ^2}u = \lambda u + h\left({\varepsilon x} \right)\,f\left(u \right),} & {x \in \mathbb{R}{^N},} \cr {\int_{\mathbb{R}{^N}} {{{\left| u \right|}^2}dx = {c^2},}} & {x \in \mathbb{R}{^N},} \cr}} \right.$$

where c, ε > 0; N ≥ 5; λ ∈ ℝ is a Lagrange multiplier and is unknown, hC(ℝN; [0;∞)); f: ℝ → ℝ is continuous function satisfying L2-subcritical growth. When ε is small enough, we get multiple normalized solutions. Moreover, we also obtain orbital stability of the solutions.

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具有 L2 次临界增长的ℝN 中非线性双谐波薛定谔方程的多重归一化解决方案
在本文中,我们考虑了以下非线性双谐波薛定谔方程的归一化解的存在性$$left\{\matrix{{Delta ^2}u = \lambda u + h\left({\varepsilon x} \right)\,f\left(u \right),} &;{x \in \mathbb{R}{^N},} \cr {int_{\mathbb{R}{^N}}{{{left| u \right|}^2}dx = {c^2},}} & {x in\mathbb{R}{^N},} \cr}}\其中 c, ε > 0; N ≥ 5; λ∈ ℝ 是拉格朗日乘数且未知,h∈ C(ℝN; [0;∞)); f: ℝ → ℝ 是满足 L2 次临界增长的连续函数。当 ε 足够小时,我们会得到多个归一化解。此外,我们还得到了解的轨道稳定性。
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