[Paper Review] On Lin-Ni's conjecture in dimensions four and six

Juncheng Wei, Bing Xu+1 more|arXiv (Cornell University)|Oct 15, 2015

Advanced Mathematical Modeling in Engineering15 references3 citations

TL;DR

This paper provides a negative answer to Lin-Ni's conjecture in four and six dimensions by constructing nontrivial solutions to the critical Neumann problem $\Delta u - \mu u + u^{\frac{n+2}{n-2}} = 0$ in smooth bounded domains $\Omega \subset \mathbb{R}^n$, for $n=4,6$, when $\mu > 0$ is sufficiently small. The construction relies on a Lyapunov-Schmidt reduction method to produce solutions that concentrate at a single interior point, proving the existence of non-constant solutions without symmetry or geometric constraints.

ABSTRACT

We give negative answers to Lin-Ni's conjecture for any four and six dimensional domains. No condition on the symmetry, geometry nor topology of the domain is needed.

Motivation & Objective

• To disprove Lin-Ni's conjecture in four and six dimensions for general smooth bounded domains without symmetry or geometric restrictions.
• To establish the existence of non-constant positive solutions to the critical Neumann problem $\Delta u - \mu u + u^{\frac{n+2}{n-2}} = 0$ in $\Omega \subset \mathbb{R}^n$ for $n=4,6$.
• To extend previous results that relied on radial symmetry or convexity to general domains by constructing solutions that concentrate at a single interior point.
• To demonstrate that the conjecture fails in dimensions four and six regardless of domain topology or geometry, using a refined Lyapunov-Schmidt reduction approach.
• To provide a quantitative energy estimate for the constructed solutions, showing their energy can be made arbitrarily large as $\mu \to 0^+$.

Proposed method

• Employing a Lyapunov-Schmidt reduction method to construct a solution $W$ that concentrates at a single interior point $Q \in \Omega$ as $\mu \to 0^+$.
• Defining a formal solution $W = U + \varepsilon^3 \hat{U} + \eta \varepsilon^3 + \text{lower order terms}$, where $U$ is the standard bubble solution $U_{1,0}(x) = \left(\frac{1}{1+|x|^2/4}\right)^{\frac{n-2}{2}}$.
• Using a cut-off function $\chi$ and a regularized Green's function $H(x,y)$ to localize the solution and control boundary effects.
• Applying asymptotic expansions and energy estimates to balance the nonlinear and linear terms in the equation $\Delta W - \varepsilon^3 W + W^{\frac{n+2}{n-2}} = 0$.
• Deriving precise estimates for the error terms $D_\Lambda S_\varepsilon[W]$, $D_{\bar{Q}} S_\varepsilon[W]$, and $D_\eta S_\varepsilon[W]$ to ensure the solution is close to a true solution.
• Computing the energy $J_\varepsilon[W]$ up to $O(\varepsilon^5)$ to verify the solution's existence and quantify its behavior as $\mu \to 0^+$.

Experimental results

Research questions

• RQ1 Does Lin-Ni's conjecture hold for general four-dimensional domains without symmetry or convexity assumptions?
• RQ2 Can non-constant solutions to the critical Neumann problem be constructed in six dimensions for small $\mu > 0$?
• RQ3 Is the failure of Lin-Ni's conjecture in dimensions four and six robust under general domain geometry and topology?
• RQ4 Can a single-point concentration solution be constructed in $\mathbb{R}^n$ for $n=4,6$ using a reduction method that avoids radial symmetry?
• RQ5 What is the asymptotic energy behavior of such non-constant solutions as $\mu \to 0^+$?

Key findings

• For any smooth bounded domain $\Omega \subset \mathbb{R}^4$ or $\mathbb{R}^6$, there exists $\mu_0 > 0$ such that for all $0 < \mu < \mu_0$, the Neumann problem $\Delta u - \mu u + u^{\frac{n+2}{n-2}} = 0$ admits a non-constant positive solution.
• The constructed solution concentrates at a single interior point $Q \in \Omega$, and its energy can be made arbitrarily large as $\mu \to 0^+$, confirming the failure of the conjecture in these dimensions.
• The energy of the solution is asymptotically given by $J_\varepsilon[W] = 4\int_{\mathbb{R}^6} U_{1,0}^3 + \left(\frac{1}{2}\eta^2|\Omega| - c_6\eta\Lambda^2 + \frac{1}{48}c_6\Lambda^2 - 8\eta^3|\Omega|\right)\varepsilon^3 + O(\varepsilon^4)$, with explicit dependence on $H(Q,Q)$ and $\Lambda$.
• The method does not require any symmetry, convexity, or mean curvature assumptions on $\Omega$, making the result valid for general domains.
• The result confirms the conjecture by Rey and Wei that Lin-Ni's conjecture fails in dimensions $n=4,5,6$, extending previous results that relied on radial or convex domains.
• The analysis in dimension six relies on precise asymptotic expansions and energy estimates up to $O(\varepsilon^5)$, with careful control of the error terms in the Lyapunov-Schmidt reduction.