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

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, h ∈ C(ℝ^N; [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.