Normalized solutions to focusing Sobolev critical biharmonic Schrödinger equation with mixed dispersion

TL;DR

This work analyzes normalized (mass-constrained) solutions to the focusing biharmonic Schrödinger equation with mixed dispersion in $\mathbb{R}^N$, $N\ge5$, including the Sobolev-critical nonlinearity. It develops a two-pronged variational strategy: in the $L^2$-subcritical range ($2<p<2+\frac{8}{N}$) it employs truncation, concentration-compactness, and genus theory to establish existence and multiplicity of normalized solutions, with a refined treatment that improves on prior results. In the $L^2$-supercritical range ($2+\frac{8}{N}<p<4^*$) it combines constrained variational methods and mountain-pass arguments, and introduces a novel dispersion-analysis component to extend recent mixed-dispersion results to the Sobolev-critical setting. A Pohozaev-manifold approach is also developed to handle the supercritical regime, yielding at least one nonnegative normalized solution with negative Lagrange multiplier for large mass. Overall, the paper broadens the landscape of normalized solutions for higher-order Schrödinger equations with mixed dispersion and critical growth, offering tools and multiplicity results applicable to related dispersive problems.

Abstract

This paper is concerned with the following focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases} Δ^2u-Δu-λu-μ|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $N \geq 5$, $μ,c>0$, $2<p<4^*:=\frac{2N}{N-4}$ and $λ\in \mathbb{R}$ 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 $Δu$. 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.

Figures (1)

• Figure 1: figure of $h_c(r)$ for $0<c<c_*$

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof