Infinitely many solutions for nonlinear fourth-order Schrödinger equations with mixed dispersion

ABSTRACT

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: $\delta {\mathrm{\Delta }}^{2}u-\mathrm{\Delta u}+u=|u{|}^{2\sigma }u,u\in H^2\left({\mathbb{R}}^N\right),$where $\delta >0$ is sufficiently small, $0<\sigma <\frac{2}{(N-2{)}_{+}}$, $\frac{2}{(N-2{)}_{+}}=\frac{2}{N-2}$ for $N\ge 3$ and $\frac{2}{(N-2{)}_{+}}=+\mathrm{\infty }$ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose $P\left(x\right)$ and $Q\left(x\right)$ are two positive, radial and continuous functions satisfying that as $r=|x|\to +\mathrm{\infty }$, $P\left(r\right)=1+\frac{a_1}{r^{m_1}}+O\left(\frac{1}{r^{m_1+{\theta }_1}}\right),Q\left(r\right)=1+\frac{a_2}{r^{m_2}}+O\left(\frac{1}{r^{m_2+{\theta }_2}}\right),$where $a_1,a_2\in \mathbb{R}$, $m_1,m_2>1$, ${\theta }_1,{\theta }_2>0$. We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: $\delta {\mathrm{\Delta }}^{2}u-\mathrm{\Delta u}+P\left(x\right)u=Q\left(x\right)|u{|}^{2\sigma }u,u\in H^2\left({\mathbb{R}}^N\right).$

KEYWORDS:

• Mixed dispersion Schrödinger equation
• polynomial decay
• Lyapunov–Schmidt reduction
• nonradial solutions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q55
• 35J30
• 35B40
• 35J50

Disclosure statement

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

Additional information

Funding

The corresponding author is partially supported by National Natural Science Foundation of China [grant number No.11901531] and China Scholarship Council [grant number 202008330417]. X. Luo is supported by National Natural Science Foundation of China [grant number s11901147 and 11771166] and the Fundamental Research Funds for the Central Universities of China [grant number JZ2020HGTB0030]. Z. Tang is supported by National Natural Science Foundation of China [grant number 12071036].