Normalized solutions to focusing Sobolev critical biharmonic Schr\"{o}dinger equation with mixed dispersion

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth, extending previous results and introducing a new method to handle dispersion effects.

Contribution

It provides new existence and multiplicity results for normalized solutions in both subcritical and supercritical cases, removing previous assumptions and developing a novel approach for mixed dispersion.

Findings

Established existence and multiplicity of solutions in the subcritical case.

Extended results to supercritical perturbation using variational methods.

Introduced a new method to handle the effects of the dispersion term.

Abstract

This paper is concerned with the following focusing biharmonic Schr\"{o}dinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases} {\Delta}^2u-\Delta u-\lambda u-\mu|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $N \geq 5$, $\mu,c>0$, $2<p<4^*:=\frac{2N}{N-4}$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier. For this problem, under the $L^2$-subcritical perturbation ($2<p<2+\frac{8}{N}$), we derive the existence and multiplicity of normalized solutions via the truncation technique, concentration-compactness principle and the genus theory presented by C.O. Alves et al. (Arxiv, (2021), doi: 2103.07940v2). Compared to the results of C.O. Alves et al. we obtain a more general result after removing the further assumptions given in (3.2) of their paper. In the case of $L^2$-supercritical perturbation ($2+\frac{8}{N}<p<4^*$), we explore the existence results of normalized solutions by applying the constrained variational methods and the mountain pass theorem. Moreover, we propose a novel method to address the effects of the dispersion term $\Delta u$. This approach allows us to extend the recent results obtained by X. Chang et al. (Arxiv, (2023), doi: 2305.00327v1) to the mixed dispersion situation.