Normalized solutions for a fourth-order Schrödinger equation with positive second-order dispersion coefficient

Xiao Luo, Tao Yang

Published 2019-08-07Version 1

We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\"{o}dinger equation $$\Delta^2u+\mu \Delta u - \lambda u= |u|^{p-2}u,\quad x \in \mathbb{R}^N\qquad (0.1)$$ under the normalized constraint $\int_{\mathbb{R}^N} u^2=a^2,$ where $N\!\geq\!2$, $a\!>\!0$, $\mu\!>\!0$, $2+\frac{8}{N}\!<\!p\!\leq\! 4^{*}\!=\!\frac{2N}{(N-4)^{+}}$ and $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier. Since the positive second-order dispersion term affects the structure of the corresponding energy functional $$E_{\mu}(u)=\frac{1}{2}{||\Delta u||}_2^2-\frac{\mu}{2}{||\nabla u||}_2^2-\frac{1}{p}{||u||}_p^p $$ we could find at least two normalized solutions to (0.1) when $2+\frac{8}{N}< p< 4^*$; at least one normalized ground state solution when $p=4^*$, under suitable assumptions on $a$ and $\mu$. Furthermore, we give some asymptotic properties of the normalized solutions to (0.1) as second-order dispersion term vanishes. In conclusion, we mainly extend the results in D. Bonheure et al. (SIAM J. Math. Anal. 2017 & Trans. Amer. Math. Soc. 2019), which deal with (0.1), from $\mu\leq0$ to the case of $\mu>0$, and also extend the results in T. Luo et al. (arXiv:1904.02540), which deal with (0.1), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.