[Paper Review] Existence and local uniqueness of bubbling solutions for poly-harmonic equations with critical growth

Yuxia Guo, Shuangjie Peng+1 more|arXiv (Cornell University)|Mar 22, 2015

Nonlinear Partial Differential Equations26 references8 citations

TL;DR

This paper establishes the existence and local uniqueness of multi-bubbling solutions for poly-harmonic equations with critical growth in $>2m+2$ dimensions, under periodic and critical-point conditions on the coefficient $K(y)$. It proves that such solutions preserve the symmetry of $K(y)$, using refined Pohozaev identity analysis and asymptotic estimates to show uniqueness in a local neighborhood of the solution sequence.

ABSTRACT

\begin{abstract} We consider the following poly-harmonic equations with critical exponents: \begin{equation}\label{P} (-Δ)^m u =K(y)u^{\frac{N+2m}{N-2m}},\;\;\; u>0\;\;\;\hbox{in} \mathbb{R}^N, \end{equation} where $N> 2m+2,m\in\mathbb{N}_{+}, K(y)$ is positive and periodic in its first $k$ variables $(y_1,\cdots, y_k)$, $1\leq k

Motivation & Objective

• To establish the existence of infinitely many bubbling solutions for poly-harmonic equations with critical exponent in dimensions $N > 2m+2$.
• To investigate the local uniqueness of these bubbling solutions, particularly under periodic and critical-point assumptions on the coefficient $K(y)$.
• To explore whether the symmetry of the prescribed scalar curvature $K(y)$ is preserved in the resulting bubbling solutions.
• To develop a refined analysis of Pohozaev identities to overcome the challenges of estimating higher-order terms in the poly-harmonic setting.
• To extend previous results on multi-bubbling solutions to the poly-harmonic case with improved regularity and symmetry conditions on $K(y)$.

Proposed method

• Constructs approximate solutions by gluing multiple bubble profiles centered at lattice points in a $k$-dimensional sublattice of $\mathbb{R}^N$, with $k < \frac{N-2m}{2}$.
• Uses a Lyapunov-Schmidt reduction method to correct approximate solutions into exact solutions, relying on the non-degeneracy of the linearized operator.
• Applies weighted $L^p$ and $L^\infty$ estimates in annular regions around each bubble center to control error terms.
• Employs a refined Pohozaev identity analysis to derive asymptotic expansions and control the behavior of solutions near concentration points.
• Implements a linearized operator analysis via the operator $A$ defined by $(AX)_j = \beta a_j^{\beta-1}x_j - \sum_i d_{ji}x_i$, proving its invertibility via spectral bounds.
• Uses the condition $K(x) = K(0) + \sum_{i=1}^N a_i |x_i|^\beta + R(x)$ with $\beta \in (N-2m, N)$ and $R(x)$ of controlled regularity to model the local structure of $K(y)$ near critical points.

Experimental results

Research questions

• RQ1 Under what conditions on $K(y)$ does the poly-harmonic equation $(-\Delta)^m u = K(y)u^{\frac{N+2m}{N-2m}}$ admit infinitely many multi-bubbling solutions in $\mathbb{R}^N$?
• RQ2 Can the local uniqueness of bubbling solutions be established in the poly-harmonic setting, despite the lack of symmetry in higher-order operators?
• RQ3 Does the symmetry of the coefficient $K(y)$, particularly its periodicity and critical-point structure, get preserved in the resulting bubbling solutions?
• RQ4 How can Pohozaev identities be adapted and estimated in the poly-harmonic case to prove uniqueness, given the complexity of higher-order derivatives?
• RQ5 What is the optimal dimension restriction $k < \frac{N-2m}{2}$, and why is it necessary for the existence and uniqueness of such solutions?

Key findings

• The paper proves the existence of infinitely many multi-bubbling solutions for the poly-harmonic equation with critical growth under the conditions $N > 2m+2$, $k < \frac{N-2m}{2}$, and $K(y)$ periodic and critical at the origin.
• It establishes the local uniqueness of the constructed bubbling solutions by showing that any two such solutions close to the same approximate solution must be identical in a small neighborhood.
• The local uniqueness result implies that the symmetry of $K(y)$—specifically its periodicity and radial-like behavior near the origin—is preserved in the bubbling solutions.
• The analysis shows that the error in the approximation decays as $O(\mu_L^{-\tau})$ for some $\tau > 0$, with $\mu_L$ being the scaling parameter of the bubbles.
• The linearized operator around the solution is shown to be invertible via the bound $\|AX\| \geq c'\|X\|$, ensuring non-degeneracy and uniqueness in the local solution space.
• The dimension restriction $k < \frac{N-2m}{2}$ is proven to be optimal, as the method fails to work beyond this threshold.