[Paper Review] Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation

Nabile Boussaïd, Antonio J. Fernández+1 more|arXiv (Cornell University)|Oct 29, 2019

Advanced Mathematical Physics Problems36 references4 citations

TL;DR

This paper investigates the existence and stability of standing wave solutions for a fourth-order nonlinear Schrödinger equation with mixed dispersion by analyzing a constrained minimization problem for the energy functional under fixed $L^2$-norm. The key contribution is proving that for small mass $c>0$, minimizers exist and are precompact up to translation when $\beta>0$, $\sigma N=4$, and $N\geq 5$, resolving the challenging vanishing case in concentration-compactness arguments.

ABSTRACT

Let $γ> 0\,$, $β> 0\,$, $α> 0$ and $0 < σN < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u), \end{equation*} where \begin{equation*} E (u):=\fracγ{2}\int_{\mathbb{R}^N}|Δu|^2\, dx -\fracβ{2}\int_{\mathbb{R}^N}| abla u|^2\, dx-\fracα{2σ+2}\int_{\mathbb{R}^N}|u|^{2σ+2}\, dx, \end{equation*} and \begin{equation*} S(c):=\left\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^{2}\, dx=c ight\}. \end{equation*} The aim of our study is twofold. On one hand, this minimization problem is related to the existence and orbital stability of standing waves for the mixed dispersion nonlinear biharmonic Schrödinger equation \begin{equation*} i \partial_t ψ-γΔ^2 ψ- βΔψ+ α|ψ|^{2σ} ψ=0, \quad ψ(0, x)=ψ_0 (x),\quad (t, x) \in \mathbb{R} imes \mathbb{R}^N. \end{equation*} On the other hand, in most of the applications of the Concentration-Compactness principle of P.-L. Lions, the difficult part is to deal with the possible dichotomy of the minimizing sequences. The problem under consideration provides an example for which, to rule out the dichotomy is rather standard while, to rule out the vanishing, here for $c > 0$ small, is challenging. We also provide, in the limit $c o 0$, a precise description of the behaviour of the minima. Finally, some extensions and open problems are proposed.

Motivation & Objective

• To establish the existence of minimizers for the energy functional constrained to fixed $L^2$-norm in the context of a biharmonic nonlinear Schrödinger equation with mixed dispersion.
• To resolve the critical challenge in applying the Concentration-Compactness principle—specifically, ruling out vanishing of minimizing sequences for small mass $c>0$ when $\beta>0$.
• To provide a precise description of the asymptotic behavior of minimizers as $c \to 0^+$, including convergence of the associated Lagrange multiplier to $\beta^2/(4\gamma)$.
• To explore the possibility of a bifurcation phenomenon from the bottom of the essential spectrum of the linear operator $\gamma\Delta^2 + \beta\Delta$.
• To extend known results on critical mass phenomena to the case $\beta>0$, where previous work focused on $\beta\leq 0$.

Proposed method

• Formulate the constrained minimization problem $m(c) = \inf_{u \in S(c)} E(u)$, where $S(c)$ is the set of $H^2$ functions with $L^2$-norm $c$, and $E(u)$ includes kinetic, potential, and nonlinear terms.
• Apply the Concentration-Compactness principle to analyze minimizing sequences, focusing on ruling out dichotomy and vanishing as $c \to 0^+$.
• Use scaling arguments and test functions of the form $w_s(x) = s^{N/2} U(sx)$, where $U$ is a ground state of the limiting problem, to analyze the behavior of $E(w_s)$ and prove coercivity for small $c$.
• Establish precompactness of minimizing sequences in $H^2(\mathbb{R}^N)$ up to translations for $c < c_N^*$, leveraging the coercivity of $E$ on $S(c)$ when $\sigma N = 4$ and $N \geq 5$.
• Derive a lower bound on the Lagrange multiplier $\lambda$ associated with minimizers, showing $\lambda > \beta^2/(4\gamma)$, which confirms the solution bifurcates from the essential spectrum.
• Use variational techniques and energy estimates to analyze the limit $c \to 0^+$, proving that minimizers converge to zero in $H^2$-norm and $\lambda \to \beta^2/(4\gamma)$.

Experimental results

Research questions

• RQ1 Under what conditions on $\gamma>0$, $\beta>0$, $\alpha>0$, $\sigma$, and $N$ does the constrained minimization problem $m(c)$ admit a minimizer for small $c>0$?
• RQ2 How can the vanishing of minimizing sequences be ruled out in the case $\beta>0$, which is more challenging than the $\beta\leq 0$ case?
• RQ3 What is the asymptotic behavior of minimizers and their associated Lagrange multipliers as $c \to 0^+$?
• RQ4 Does a bifurcation phenomenon occur at the bottom of the essential spectrum $\lambda = \beta^2/(4\gamma)$ for the nonlinear equation?
• RQ5 Can the existence of minimizers be extended to the mass-critical case $\sigma N = 4$ when $\beta>0$?

Key findings

• For $\beta>0$, $\sigma N = 4$, and $N \geq 5$, the functional $E$ is coercive on $S(c)$ for all $c < c_N^*$, ensuring the existence of minimizers for small $c>0$.
• Any minimizing sequence for $m(c)$ with $c < c_N^*$ is precompact in $H^2(\mathbb{R}^N)$ up to translations, which rules out vanishing and dichotomy.
• The minimizer $u$ of $m(c)$ satisfies $\lambda > \beta^2/(4\gamma)$, confirming that the solution bifurcates from the essential spectrum of the linear operator.
• As $c \to 0^+$, any minimizer $u_c$ converges to zero in $H^2(\mathbb{R}^N)$-norm, and the associated Lagrange multiplier $\lambda_c$ converges to $\beta^2/(4\gamma)$.
• The critical mass threshold $\tilde{c}$, which exists for $\beta<0$, is zero when $\beta>0$, indicating that minimizers exist for all small $c>0$.
• The paper conjectures that a bifurcation phenomenon from the essential spectrum occurs not only for $\sigma N < 4$ but also for $\sigma \in (4/N, 1)$ when $\sigma N > 4$, based on analogy with previous results.