Multiplicity of normalized solutions to biharmonic Schrödinger equation with mixed nonlinearities

Abstract

This paper is concerned with the existence of multiple normalized solutions to the elliptic problems $\{\begin{array}{ll}{\mathrm{\Delta }}^{2}u=\mathrm{\lambda u}+h\left(\mathrm{ϵx}\right)|u{|}^{q-2}u+|u{|}^{p-2}u& \text{in}{\mathbb{R}}^N,\\ {\displaystyle {\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=c,}& \end{array}$ where c, $ϵ>0$, $N\ge 5$, $2<q<2+\frac{8}{N}<p\le 4^{\ast }:=\frac{2N}{N-4}$, $\lambda \in \mathbb{R}$ is a Lagrangian multiplier and $h:{\mathbb{R}}^N\to [0,\mathrm{\infty })$ is a continuous function. We find some $c_0>0$ and present a class of reasonable assumptions on h to guarantee that the numbers of normalized solutions are at least the numbers of global maximus points of h when ε is small enough and $0<c\le c_0$.

Keywords:

• Nonlinear biharmonic Schrödinger equation
• normalized solutions
• multiplicity
• variational methods

AMS Subject Classifications:

• 35A15
• 35J35
• 35J30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

The authors declare that data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.