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

Abstract

We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schrödinger equation \begin{equation}\label{1} Δ^{2}u+μΔu-λu={|u|}^{p-2}u, ~~~~x \in \R^{N}\\ \end{equation} under the normalized constraint $${\int }_{{\mathbb{R}}^N}{u}^2=a^2,$$ where $N\ge 2$, $a,\mu >0$, $2+\frac{8}{N}<p<4^{\ast }=\frac{2N}{(N-4{)}^{+}}$ and $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. Since the 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 (\ref{1}) if $2+\frac{8}{N}<p<4^{\ast }$ and ${\mu }^{p{\gamma }_p-2}{a}^{p-2}<C$ for some explicit constant $C=C(N,p)>0$ and ${\gamma }_p=\frac{N(p-2)}{4p}$. Furthermore, we give some asymptotic properties of the normalized solutions to (\ref{1}) as $\mu \to 0^{+}$ and $a\to 0^{+}$, respectively. In conclusion, we mainly extend the results in \cite{DBon,dbJB}, which deal with (\ref{1}), from $\mu \le 0$ to the case of $\mu >0$, and also extend the results in \cite{TJLu,Nbal}, which deal with (\ref{1}), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.

arXiv commentarXiv admin note: This is a revised version of the previous one. text overlap with arXiv:1811.00826, arXiv:1901.02003 by other authors

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• Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation