Compactness results for Sign-Changing Solutions of critical nonlinear elliptic equations of low energy

1. Problem Statement

The paper investigates the compactness properties of the set of sign-changing solutions to the critical nonlinear elliptic equation with a potential term $h$:
$$\begin{cases} -\Delta v + hv = |v|^{2^*-2}v & \text{in } \Omega, \\ v = 0 & \text{on } \partial\Omega, \end{cases}$$
where:

• $\Omega \subset \mathbb{R}^n$ ($n \ge 3$) is a smooth, bounded, connected domain.
• $2^* = \frac{2n}{n-2}$ is the critical Sobolev exponent.
• $h \in C^1(\Omega)$ is a potential function.
• The operator $-\Delta + h$ is assumed to be coercive.

The central question is whether sequences of solutions $(v_\alpha)$ with minimal sign-changing energy (specifically, energy bounded by the threshold $K_n^{-n} + I_h(\Omega)^{n/2}$, where $K_n$ is the Sobolev constant and $I_h$ is the ground state energy) are compact in the $C^2(\Omega)$ topology.

Context:

• For positive solutions, compactness is well-understood but fails in certain dimensions or energy regimes (e.g., the Brézis-Nirenberg problem).
• For sign-changing solutions, the behavior is generally more erratic. While existence results exist, compactness at the lowest energy level for sign-changing solutions had not been fully established, particularly regarding the interaction between the concentration of bubbles and the boundary of the domain.

2. Methodology

The authors employ a blow-up analysis combined with contradiction arguments. The core strategy involves:

1. Assumption of Non-Compactness: Assume a sequence of sign-changing solutions $(v_\alpha)$ blows up (i.e., $\|v_\alpha\|_\infty \to \infty$) while satisfying the energy bound.
2. Decomposition: Utilizing Struwe's global compactness result, the sequence is decomposed as:
   $$v_\alpha = B_\alpha \pm v_\infty + o(1) \quad \text{in } H^1_0(\Omega),$$
   where $B_\alpha$ is a positive "bubble" concentrating at $x_\alpha$ with scale $\mu_\alpha$, and $v_\infty$ is a non-negative solution (the weak limit).
3. New Pointwise Estimates (The Core Innovation):
   • The authors derive a global pointwise description of the blowing-up sequence that holds up to the boundary $\partial\Omega$.
   • They prove that:
     $$\left| \frac{v_\alpha - \Pi B_\alpha \mp v_\infty}{B_\alpha + v_\infty} \right|_\infty \to 0,$$
     where $\Pi B_\alpha$ is the projection of the bubble $B_\alpha$ onto $H^1_0(\Omega)$ (solving the equation with the bubble's nonlinearity as a source).
   • This estimate is crucial because, near the boundary, $B_\alpha$ and $\Pi B_\alpha$ behave differently, and $v_\infty$ vanishes. Previous estimates for positive solutions did not handle sign-changing solutions near the boundary with this precision.
4. Pohozaev Identities:
   • The authors apply Pohozaev identities at different scales (interior scales $\sqrt{\mu_\alpha}$ and boundary scales related to the distance $d_\alpha = \text{dist}(x_\alpha, \partial\Omega)$).
   • They analyze the asymptotic behavior of these identities to derive necessary conditions on the potential $h$ and the limit solution $v_\infty$ for a blow-up to occur.
5. Case Analysis:
   • Interior Blow-up ($x_\alpha \to x_\infty \in \Omega$): Using Pohozaev identities, they derive constraints on $h(x_\infty)$ and the mass function $m_h(x_\infty)$.
   • Boundary Blow-up ($x_\alpha \to x_\infty \in \partial\Omega$): This is the most novel and difficult part. They introduce a specific scaling $r_\alpha = \sqrt{\mu_\alpha} / d_\alpha^{1/(n-2)}$ to analyze the interaction between the bubble and the boundary. They prove that for sign-changing solutions, boundary blow-up leads to contradictions under specific dimensional and potential constraints.

3. Key Contributions

• First Global Pointwise Estimate for Sign-Changing Solutions: The paper establishes a sharp pointwise estimate (Theorem 2.1) for sign-changing blowing-up solutions that is valid up to the boundary. This is a significant technical advancement over existing literature which focused primarily on positive solutions or interior blow-up.
• Resolution of Boundary Concentration for Sign-Changing Solutions: The authors prove that under certain conditions, sign-changing solutions cannot concentrate at the boundary. This contrasts with positive solutions (where boundary concentration is impossible for $n \ge 4$) and highlights the unique, complex behavior of sign-changing solutions.
• Dimensional Dichotomy: The results clarify the role of dimension $n$:
  • Low Dimensions ($3 \le n \le 5$): Compactness holds unconditionally for sign-changing solutions at the lowest energy level.
  • High Dimensions ($n \ge 7$): Compactness holds provided $h \neq 0$ everywhere.
  • Critical Dimension ($n=6$): Compactness fails; the paper characterizes the specific non-compact behavior (Proposition 1.1).

4. Main Results

Theorem 1.1 (Main Compactness Result):
Let $(v_\alpha)$ be a sequence of sign-changing solutions with energy bounded by $K_n^{-n} + I_h(\Omega)^{n/2}$. Then $(v_\alpha)$ is compact in $C^2(\Omega)$ (up to a subsequence) if:

• Case A: $n \in \{3, 4, 5\}$. (No extra condition on $h$).
• Case B: $n \ge 7$ and $h(x) \neq 0$ for all $x \in \Omega$.

Proposition 1.1 (The $n=6$ Case):
In dimension $n=6$, compactness fails. If a sequence blows up, the limit $v_\infty$ is a positive solution, and the concentration point $x_\infty$ satisfies the specific relation:
$$h(x_\infty) = \pm 2v_\infty(x_\infty).$$
This recovers known non-compactness phenomena for positive solutions but extends the characterization to the sign-changing context.

Corollary 1.1:
The set of sign-changing solutions satisfying the energy bound is precompact in $C^2(\Omega)$ under the conditions of Theorem 1.1.

5. Significance

• Completeness of Theory: This work fills a major gap in the theory of critical elliptic equations. While positive solutions were extensively studied (e.g., Brézis-Nirenberg, Druet), the compactness of sign-changing solutions at the lowest energy level was an open problem.
• Novelty of Boundary Analysis: The proof that sign-changing solutions cannot concentrate at the boundary in low dimensions (or when $h \neq 0$ in high dimensions) is a profound result. It demonstrates that the "sign-changing" nature imposes stricter constraints on blow-up locations than the "positive" nature does in certain regimes.
• Technical Framework: The new pointwise estimates (Theorem 2.1) provide a robust tool for future research on sign-changing solutions, particularly for problems involving boundaries or perturbations of the potential $h$.
• Counter-Examples and Sharpness: The paper explicitly identifies $n=6$ as the dimension where compactness breaks down and provides the precise asymptotic behavior, showing the sharpness of their results. It also notes that if the energy bound is relaxed, non-compactness can occur even in low dimensions (Remark 3.3), suggesting the energy threshold is optimal.

In summary, this paper provides a definitive answer to the compactness of low-energy sign-changing solutions, introducing powerful new analytical tools to handle the interaction between sign-changing profiles and domain boundaries.