Power law convergence and concavity for the Logarithmic Schrödinger equation

1. Problem Statement

The paper investigates the Logarithmic Schrödinger equation with Dirichlet boundary conditions in a bounded, convex domain $\Omega \subset \mathbb{R}^N$:
$$\begin{cases} -\Delta u = u \log u^2 & \text{in } \Omega \\ u > 0 & \text{in } \Omega \\ u = 0 & \text{on } \partial\Omega \end{cases} \quad (1.1)$$
The primary goal is to establish concavity properties of positive solutions. Specifically, the authors aim to prove the existence of a solution $u$ such that $\log u$ is concave (i.e., $u$ is log-concave).

Context and Challenges:

• Superlinear, Sign-Changing Source: The reaction term $f(u) = u \log u^2$ is superlinear but changes sign (negative for $u < 1$, positive for $u > 1$). This prevents the direct application of classical concavity maximum principles (e.g., Korevaar's method) which typically require non-negative or specific monotonicity properties of the reaction.
• Lack of Uniqueness: Unlike sublinear problems, the uniqueness of ground states for superlinear equations in general convex domains is not guaranteed, complicating continuity arguments.
• Nondispersive Nature: The equation arises in physics (atomic physics, quantum hydrodynamics) and represents a nondispersive wave equation, distinguishing it from the more standard dispersive cases.

2. Methodology

The authors employ a power-law approximation strategy, bridging the logarithmic equation with the classical Lane-Emden equation:
$$\begin{cases} -\Delta u = \sigma (u^q - u) & \text{in } \Omega \\ u > 0 & \text{in } \Omega \\ u = 0 & \text{on } \partial\Omega \end{cases} \quad (1.2)$$
The core strategy involves three main pillars:

A. Power Law Approximation ($q \to 1^+$)

The authors utilize the heuristic limit:
$$\lim_{q \to 1^+} \frac{2}{q-1}(u^q - u) = u \log u^2$$
They set $\sigma_q = \frac{2}{q-1}$ and study the asymptotic behavior of solutions $u_q$ to (1.2) as $q \to 1^+$. They prove that these solutions converge to a solution of the Logarithmic Schrödinger equation (1.1).

B. Concavity via Constant Rank Theorem

To establish concavity for the approximating Lane-Emden solutions, they use the Constant Rank Theorem (Caffarelli-Friedman, Korevaar-Lewis).

• They analyze the transformation $w = \phi(u) = u^{(1-q)/2}$.
• They verify conditions ensuring that the Hessian $D^2 w$ has constant rank and is positive definite (strong convexity), which implies $u^{(1-q)/2}$ is convex, or equivalently, $u$ is $(1-q)/2$-concave.
• A crucial step involves proving that the rank is full (strong convexity) even in domains that are not strictly smooth, using a Liouville theorem on convex epigraphs.

C. Liouville Theorems on Convex Epigraphs

To handle general convex domains (which may have corners or flat parts) and perform blow-up arguments, the authors prove a new Liouville theorem for the equation $-\Delta v = v^q$ in a convex epigraph $H$ (a domain defined as the set above the graph of a convex function).

• They show that no bounded positive solutions exist in such domains for certain ranges of $q$.
• This result is essential for deriving uniform a-priori bounds on solutions that are independent of the smoothness of $\partial\Omega$.

3. Key Contributions and Results

Theorem 1.1: Existence of a Log-Concave Solution

Result: For any bounded, convex domain $\Omega$, there exists a locally strongly log-concave solution to the Logarithmic Schrödinger equation (1.1).
Significance: This is one of the few results establishing log-concavity for a superlinear, sign-changing source term. It confirms that the solution's super-level sets are strictly convex.

Theorem 1.2: Concavity for Lane-Emden Equation

Result: For the Lane-Emden equation (1.2) with $\sigma > 0$ and $q \in (1, 2^*-1)$, there exists a solution $u$ such that $u^{(1-q)/2}$ is strongly convex (i.e., $u$ is $(1-q)/2$-concave).
Significance: This extends known concavity results (previously limited to specific domains or $q$ near 1) to general convex domains and a wide range of $q$.

Theorem 1.3: Convergence of Solutions

Result: As $q \to 1^+$ with $\sigma_q = 2/(q-1)$, solutions $u_q$ of (1.2) converge (up to subsequences) in $C^0(\Omega) \cap C^2_{loc}(\Omega) \cap W^{1,2}_0(\Omega)$ to a solution of (1.1). If $u_q$ are ground states, the limit is a ground state.
Significance: This rigorously justifies the power-law approximation method for the logarithmic equation.

Theorem 1.4: Liouville Theorem on Convex Epigraphs

Result: In a convex epigraph $H$, the equation $-\Delta u = u^q$ ($q \ge 1$) admits no bounded positive solutions if $q \le 2^*-1$. Furthermore, any bounded solution must satisfy a monotonicity condition in a specific direction.
Significance: This is a novel geometric analysis result that allows the authors to bypass smoothness assumptions on the domain boundary during blow-up arguments.

Theorem 1.5: Dispersive Case (Allen-Cahn type)

Result: For the equation with the opposite sign ($-\Delta u = -u \log u^2$), the solution is unique and strictly quasi-concave. Moreover, specific transformations (like $\sqrt{1-\log u^2}$) are strongly convex.
Significance: Highlights the contrast between the nondispersive (harder) and dispersive (easier) cases.

4. Technical Innovations

1. Handling Non-Smooth Domains: The authors do not assume $\partial\Omega$ is $C^2$. They approximate general convex domains with smooth, strongly convex ones. The robustness of their a-priori estimates (derived via the Liouville theorem on epigraphs) allows them to pass to the limit without losing the concavity property.
2. Connectedness of Solution Branches: Since uniqueness of ground states for (1.2) is not known for general $q$, they use degree theory to construct a connected branch of solutions. This allows them to use a continuity argument starting from $q \approx 1$ (where uniqueness and log-concavity are known) to extend the result to all $q > 1$.
3. Optimality Analysis (Section 6):
   • They prove that log-concavity is the strongest power concavity ($\alpha$-concavity) generally expected for (1.1) in arbitrary convex domains.
   • They construct examples (using tensorization) showing that for any $\alpha \in (0, 1/N]$, there exists a domain and a solution that is $\alpha$-concave but not $\beta$-concave for any $\beta > \alpha$.
   • In 1D, they show solutions are actually $1/2$-log-concave (related to the function $\sqrt{-\log u}$), which is stronger than standard log-concavity.

5. Significance and Impact

• Mathematical Physics: The Logarithmic Schrödinger equation models Bose-Einstein condensates and nonlinear optics. Understanding the geometric properties (concavity) of solutions is crucial for stability analysis and physical interpretation.
• PDE Theory: The paper resolves a gap in the theory of concavity for superlinear problems with sign-changing sources. It demonstrates that the "heuristic" limit from power laws to logarithmic laws preserves deep geometric properties like log-concavity.
• Geometric Analysis: The development of the Liouville theorem for convex epigraphs provides a new tool for studying elliptic equations in non-smooth or unbounded domains, potentially applicable to other frameworks involving convex geometry.
• Methodological Advancement: The combination of degree theory, constant rank theorems, and blow-up analysis on non-smooth domains offers a robust template for tackling similar nonlinear elliptic problems where standard uniqueness or regularity assumptions fail.

In summary, the paper successfully establishes the log-concavity of solutions to the Logarithmic Schrödinger equation in general convex domains by rigorously linking it to the concavity properties of the Lane-Emden equation via a power-law limit, supported by novel geometric analysis tools.