On the fractional logarithmic $p$-Laplacian

Technical Summary: On the Fractional Logarithmic p-Laplacian

Problem Statement and Motivation
This paper introduces and investigates the fractional logarithmic p-Laplacian, denoted as $(-\Delta)^{s+\log}_p$. The operator is defined as the first-order derivative with respect to the order parameter $t$ of the fractional p-Laplacian $(-\Delta)^t_p$, evaluated at an arbitrary point $s \in (0, 1)$. While the linear case ($p=2$) and the limit case $s \to 0$ (leading to the logarithmic Laplacian $L_\Delta$) have been recently studied by Chen et al. and Dyda et al., this work extends the theory to the nonlinear, quasilinear setting where $1 < p < \infty$.

The primary motivation stems from the need to understand the asymptotic behavior of fractional operators and to develop a rigorous functional framework for problems driven by this new nonlocal operator. A significant challenge addressed is the differentiation of the nonlocal integral kernel under the integral sign, which introduces a logarithmic singularity $|x-y|^{-N-sp} \ln|x-y|$. This logarithmic factor alters the singularity structure, particularly in the range $1 < p < 2$, and changes the sign of the kernel for $|x-y| > 1$, complicating standard variational arguments regarding coercivity and positivity.

Methodology and Functional Framework
The authors establish a comprehensive functional analytic framework based entirely on real-variable methods, avoiding Fourier transform techniques which are unavailable for $p \neq 2$.

1. Operator Definition and Representation:
   The paper proves that for $u \in C^2_c(\mathbb{R}^N)$, the mapping $t \mapsto (-\Delta)^t_p u(x)$ is $C^1$ on $(0, 1)$. The derivative yields the explicit integral representation:
   $$(-\Delta)^{s+\log}_p u(x) = B(N, s, p)(-\Delta)^s_p u(x) - p C(N, s, p) \text{P.V.} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)) \ln|x-y|}{|x-y|^{N+sp}} dy$$
   where $C(N, s, p)$ is a specific normalization constant ensuring correct limits as $s \to 0$ and $s \to 1$, and $B(N, s, p) = \frac{d}{ds} \ln C(N, s, p)$. The kernel $K^{s+\log}_p$ is shown to be sign-changing, decomposing into positive and negative parts.

2. Energy Spaces:
   The natural energy space $W^{s+\log,p}_0(\Omega)$ is introduced, defined via a Gagliardo-type seminorm involving the positive part of the logarithmic kernel. The authors prove this space is a separable, uniformly convex, and reflexive Banach space. They establish that this space sits strictly between standard fractional Sobolev spaces $W^{s+\epsilon,p}_0(\Omega)$ and $W^{s,p}_0(\Omega)$.

3. Key Inequalities:
   The paper derives several fundamental inequalities essential for analysis:

   • Poincaré-type inequality: Establishing control of the $L^p$ norm by the fractional logarithmic seminorm.
   • Hardy-type inequality: A boundary Hardy inequality with a logarithmic weight near the boundary.
   • Díaz-Saa inequality: Proven under the geometric condition $\text{diam}(\Omega) < e^{-1/sp}$, leading to uniqueness results.
   • Gagliardo-Nirenberg inequality: Interpolating between $L^p$ and the fractional logarithmic norm.
   • Pohozaev Identity: A modified Pohozaev identity is derived, revealing a strictly positive "defect measure" $\Gamma_{s,p}(u)$ arising from the logarithmic weight breaking the homogeneity of the classical kernel.

Key Results

• Embedding Properties: The paper establishes continuous embeddings of $W^{s+\log,p}_0(\Omega)$ into $L^q(\Omega)$ for $1 \le q \le p^*_s$ (where $p^*_s = Np/(N-sp)$). Crucially, the authors prove compact embedding at the critical exponent $p^*_s$. This distinguishes the fractional logarithmic setting from classical fractional Sobolev spaces, where compactness fails at the critical exponent. This result implies that Dirac mass concentrations do not form, rendering concentration-compactness arguments unnecessary on bounded domains.

• Dirichlet Eigenvalue Problem: The authors investigate the nonlinear eigenvalue problem:
  $$(-\Delta)^{s+\log}_p u = \lambda |u|^{p-2}u \quad \text{in } \Omega, \quad u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega.$$
  Under the geometric assumption $\text{diam}(\Omega) < e^{-1/sp}$ (which ensures the kernel remains strictly positive on $\Omega \times \Omega$) and $B(N, s, p) \ge 0$, they prove:

  1. Existence: The first eigenvalue $\lambda^{s+\log}_{1,p}$ is attained via the direct method of calculus of variations.
  2. Positivity and Simplicity: The first eigenvalue is strictly positive, and the associated eigenfunction $\phi^{s+\log}_{1,p}$ is simple (unique up to a scalar multiple) and strictly positive almost everywhere in $\Omega$.
  3. Boundedness: Eigenfunctions are globally bounded ($L^\infty(\Omega)$).
  4. Strong Minimum Principle: A strong minimum principle is established for weak supersolutions, extending previous results for fractional operators.

• Pohozaev Non-Existence vs. Existence: The derived Pohozaev identity includes a defect term $\Gamma_{s,p}(u) > 0$. The authors note that unlike the classical fractional p-Laplacian, where the Pohozaev identity often leads to non-existence results for critical exponents on star-shaped domains, the presence of the positive defect measure in the logarithmic case allows for the possibility of critical and supercritical solutions.

Significance and Claims
The paper claims to provide the first rigorous variational framework for the fractional logarithmic p-Laplacian in the nonlinear regime ($p \neq 2$). Its significance lies in:

1. Extending Linear Theory: Successfully generalizing the linear logarithmic Laplacian theory to the quasilinear setting, overcoming the difficulties posed by the lack of Fourier multipliers.
2. Critical Compactness: Demonstrating that the logarithmic correction induces a stronger compactness property at the critical Sobolev exponent, a feature not present in standard fractional Sobolev spaces.
3. Structural Insights: Revealing how the logarithmic weight breaks scale invariance, leading to a modified Pohozaev identity with a defect measure that fundamentally alters the existence theory for critical problems.
4. Spectral Properties: Establishing the existence, simplicity, and positivity of the first eigenvalue under specific geometric constraints, providing a foundation for further spectral analysis of nonlocal operators with logarithmic singularities.

The work is presented as a foundational step, developing the necessary tools (functional spaces, inequalities, and regularity estimates) to study boundary value problems driven by this operator, without proposing specific physical applications beyond the mathematical context of nonlocal phenomena.