A general theory for the $(s, p)$-superposition of nonlinear fractional operators

1. Problem Statement

The paper addresses the existence and properties of solutions to partial differential equations involving the continuous superposition of fractional $p$-Laplacian operators with varying orders of differentiation ($s$) and nonlinearity exponents ($p$).

The central operator is defined as:
$$L_\mu u := \iint_{\Sigma} (-\Delta)^s_p u \, d\mu(s, p)$$
where:

• $\Sigma = [0, 1] \times (1, N)$ is the parameter space for the fractional order $s$ and the exponent $p$.
• $\mu$ is a signed measure on $\Sigma$, decomposed as $\mu = \mu^+ - \mu^-$.
• The equation is considered under Dirichlet boundary conditions ($u=0$ in $\mathbb{R}^N \setminus \Omega$).

Key Novelty: Previous literature focused on superpositions in either $s$ (fixed $p$) or $p$ (fixed $s$), or required $\mu$ to be a positive measure. This work introduces a framework where the superposition occurs simultaneously in both $s$ and $p$, and crucially, allows for signed measures (operators with "wrong" signs, which can model phenomena like chemotaxis or inverted time diffusion).

The paper investigates two main problem types:

1. Linear/Inhomogeneous Case: $L_\mu u = g(x)$, where $g$ is a given source term.
2. Nonlinear Case: $L_\mu u = f(x, u)$, where $f$ satisfies subcritical growth conditions.

2. Methodology and Functional Framework

Functional Space Construction

The authors define a new Banach space $X_\mu(\Omega)$ tailored to the measure $\mu$.

• Norm Definition: Instead of integrating the $p$-th power of the Gagliardo seminorms (which lacks homogeneity), they define the norm as the integral of the seminorms themselves:
  $$\|u\|_\mu := \iint_{\Sigma} [u]_{s,p} \, d\mu^+(s, p)$$
  where $[u]_{s,p}$ is the standard Gagliardo seminorm (or $L^p$ norm if $s=0$, or gradient norm if $s=1$).
• Completeness: $X_\mu(\Omega)$ is defined as the completion of $C_0^\infty(\Omega)$ with respect to this norm.
• Embedding: The authors prove that $X_\mu(\Omega)$ is continuously and compactly embedded into appropriate Lebesgue and Sobolev spaces, depending on the "largest" support of the positive measure $\mu^+$. Specifically, if $\mu^+$ has support near $(s_\sharp, p_\sharp)$, the space embeds into $W^{s_\sharp, p_\sharp}_0(\Omega)$ or $L^{(p_\sharp)^*_{s_\sharp}}(\Omega)$.

Handling Signed Measures (The "Reabsorbing" Property)

A major technical hurdle is the presence of the negative part of the measure, $\mu^-$. The operator $L_\mu$ involves a subtraction of terms, which could destroy coercivity.

• Assumption: The authors assume $\mu^-$ is supported on "smaller" fractional orders (specifically $s < s_{max}$) and is quantitatively dominated by $\mu^+$.
• Reabsorption: They prove that under specific quantitative conditions (involving a small parameter $\gamma$), the negative contribution can be "reabsorbed" by the positive part. This ensures the energy functional remains coercive and bounded from below.

Variational Approach

• Case 1 (Global Minimizer): For the linear problem with a source term $g$, the authors construct an energy functional $J(u)$. They prove $J$ is weakly lower semicontinuous and coercive (under the smallness condition on $\mu^-$). By the Weierstrass Theorem, a global minimizer exists, which corresponds to a weak solution.
• Case 2 (Mountain Pass): For the nonlinear problem with $f(x,u)$, they construct a functional $I(u)$. They verify the Ambrosetti-Rabinowitz condition and the geometry required for the Mountain Pass Theorem. They establish the Palais-Smale condition by proving that bounded Palais-Smale sequences converge strongly, utilizing the uniform convexity of the space (in the case of discrete measures) and compact embeddings.

3. Key Contributions

1. Unified Framework for $(s, p)$-Superposition: The paper provides the first general theory for operators superposed in both the fractional order and the nonlinearity exponent simultaneously.
2. Signed Measures: It rigorously handles operators with negative coefficients (signed measures), showing that solutions exist even when the operator is not strictly positive definite, provided the negative part is "small" or supported on lower-order terms.
3. New Functional Space: The introduction of the space $X_\mu(\Omega)$ with the specific norm $\|u\|_\mu = \int [u]_{s,p} d\mu^+$ is a novel construction that avoids the homogeneity issues of standard $L^p$-type integrals over measures.
4. Uniform Convexity: The authors investigate the geometric properties of $X_\mu(\Omega)$, proving it is uniformly convex when $\mu^+$ is a convergent series of Dirac measures.

4. Main Results

Theorem 1.1 (Existence for Linear/Inhomogeneous Problems)

• Context: Problem $L_\mu u = g(x)$ with $\mu$ being a signed measure.
• Condition: The negative part $\mu^-$ must be supported on $s < s_{max}$ and satisfy a smallness condition relative to $\mu^+$ (specifically $\mu^-_s([0, s)) \leq \gamma \mu^+_s([s, 1])$ for small $\gamma$).
• Result: There exists a weak solution which is a global minimizer of the associated energy functional. If $\mu^- \equiv 0$, the solution is unique.

Theorem 1.5 (Existence for Nonlinear Problems)

• Context: Problem $L_\mu u = f(x, u)$ where $\mu$ is a positive measure (sum of Dirac masses).
• Conditions: $f$ satisfies subcritical growth, vanishes at zero, and satisfies the Ambrosetti-Rabinowitz condition.
• Result: There exists a nontrivial weak solution of Mountain Pass type.

Corollaries and Applications

The generality of the main theorems yields several new specific results:

• Finite Sums: Existence for sums of different fractional $p$-Laplacians (e.g., $(-\Delta)^{s_1}_{p_1} + (-\Delta)^{s_2}_{p_2}$).
• Mixed Operators: Existence for combinations like $(-\Delta)^s_p + (-\Delta)^1_p$ (fractional + standard).
• Continuous Superpositions: Existence for integrals over $s$ or $p$ (e.g., $\int \omega(s)(-\Delta)^s_p u \, ds$).
• "Wrong Sign" Operators: Existence for operators like $(-\Delta)^{s_1}_p - \alpha (-\Delta)^{s_2}_p$ provided $\alpha$ is small enough.

5. Significance

• Mathematical Rigor: The paper resolves the well-posedness issues associated with superposing operators of different orders and types, providing a robust variational framework where previous methods failed.
• Modeling Capabilities: By allowing signed measures, the theory opens the door to modeling complex physical and biological phenomena where diffusion processes might have competing mechanisms (e.g., standard diffusion vs. aggregation/chemotaxis modeled by "negative" diffusion).
• Flexibility: The framework is broad enough to encompass discrete sums, continuous integrals, and mixed scenarios, unifying various scattered results in the literature into a single coherent theory.
• Open Problems: The authors explicitly note that the case of a general measure (not just a sum of Dirac masses) satisfying the positivity condition remains an open problem, particularly regarding the uniform convexity of the space, inviting future research.

In summary, this paper establishes a foundational theory for a broad class of nonlocal, nonlinear, mixed-order operators, proving existence and uniqueness results through innovative functional analytic techniques and variational methods.