An existence theory for superposition operators of mixed order subject to jumping nonlinearities

1. Problem Statement

The paper investigates the existence of nontrivial solutions for a critical elliptic boundary value problem involving a superposition operator of mixed fractional orders coupled with a jumping nonlinearity.

The specific problem is formulated as:
$$\begin{cases} \int_{[0,1]} (-\Delta)^s u \, d\mu(s) = b u^+ - a u^- + |u|^{2^*_{s_\sharp}-2}u & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb{R}^N \setminus \Omega, \end{cases}$$
where:

• $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a bounded open set.
• $(-\Delta)^s$ denotes the fractional Laplacian of order $s \in (0,1)$.
• $\mu = \mu^+ - \mu^-$ is a signed measure on $[0,1]$, allowing for a linear combination of fractional operators with potentially different signs (including "wrong" signs).
• $u^+ = \max\{u, 0\}$ and $u^- = \max\{-u, 0\}$.
• $a, b > 0$ are parameters. The term $b u^+ - a u^-$ represents a jumping nonlinearity (non-differentiable at $u=0$).
• $2^*_{s_\sharp} = \frac{2N}{N-2s_\sharp}$ is the critical fractional Sobolev exponent, where $s_\sharp$ is a specific parameter derived from the support of the measure $\mu$.

The core challenge lies in handling the critical growth (loss of compactness in Sobolev embeddings) combined with the non-differentiability of the jumping nonlinearity and the complexity introduced by the signed measure (operators with competing diffusive trends).

2. Methodology

The authors employ a variational approach combined with spectral analysis and careful energy estimates.

A. Functional Setting and Operator Definition

• Space: They define a Hilbert space $X(\Omega)$ consisting of functions vanishing outside $\Omega$ with finite energy relative to the positive part of the measure $\mu^+$.
• Signed Measure Handling: A crucial technical innovation is the treatment of the negative part of the measure ($\mu^-$). The authors assume that the "negative" operators (which could destabilize the problem) are dominated by the "positive" operators on higher exponents. Specifically, they assume there exists $s \in (0,1]$ such that $\mu^-$ is supported on $[0,s]$ and $\mu^+$ is supported on $[s,1]$, with a quantitative bound:
  $$\mu^-([0,s]) \leq \gamma \mu^+([s,1])$$
  where $\gamma$ is sufficiently small.
• Absorption Lemma: Using fractional Sobolev inequalities, they prove that the energy contribution from the negative part of the measure can be "reabsorbed" into the positive part if $\gamma$ is small enough. This ensures the coercivity of the associated energy functional.

B. Variational Framework

• Energy Functional: The problem is cast as finding critical points of the functional $E: X(\Omega) \to \mathbb{R}$:
  $$E(u) = \frac{1}{2} \int_{[0,1]} [u]_s^2 d\mu^+(s) - \frac{1}{2} \int_{[0,s]} [u]_s^2 d\mu^-(s) - \frac{1}{2} \int_\Omega (a(u^-)^2 + b(u^+)^2) dx - \frac{1}{2^*_{s_\sharp}} \int_\Omega |u|^{2^*_{s_\sharp}} dx$$
  where $[u]_s$ is the Gagliardo seminorm.
• Dancer-Fučík Spectrum: The authors utilize the geometry of the Dancer-Fučík spectrum of the operator $A_\mu$. They identify regions in the $(a,b)$ parameter plane (specifically below the lower curve of the spectrum in a specific square $Q_l$) where the functional satisfies the geometric conditions required for existence (Mountain Pass geometry).

C. Compactness and Critical Levels

• Palais-Smale Condition: Due to the critical exponent, the functional does not satisfy the Palais-Smale (PS) condition globally. The authors establish a local PS condition below a specific energy threshold $c^*$.
• Sobolev Constant: They define a generalized Sobolev constant $S$ associated with the operator $A_\mu$.
• Threshold Calculation: By analyzing the energy levels of the functional on eigenspaces, they determine a critical energy level $c^*$ (dependent on $S$, $|\Omega|$, and the spectral gap). They prove that if the parameters $(a,b)$ satisfy a specific inequality involving the eigenvalues $\lambda_l$ and $S$, the PS condition holds for levels $c < c^*$.

3. Key Contributions

1. General Superposition Operators: The paper provides a unified existence theory for operators formed by the continuous or discrete superposition of fractional Laplacians of different orders.
2. Signed Measures ("Wrong Sign" Operators): A major novelty is the ability to handle signed measures where the operator includes terms with "wrong" signs (e.g., $-\Delta - \alpha(-\Delta)^s$). The authors show that if the "negative" diffusion is sufficiently small relative to the "positive" diffusion, existence results still hold. This models physical scenarios with competing diffusive and concentrating mechanisms.
3. Jumping Nonlinearities in Critical Settings: The work extends the study of non-differentiable (jumping) nonlinearities to the critical case for mixed-order operators, a setting previously unexplored for such general operators.
4. Dimensional Improvement: For the classical Laplacian case ($\mu = \delta_1$), the result improves upon previous literature by establishing existence for $N \geq 3$, whereas earlier results often required $N \geq 4$.

4. Main Results

Theorem 1.1 (Main Existence Theorem):
Let $\mu = \mu^+ - \mu^-$ satisfy the dominance conditions (1.3)-(1.5). Let $(a,b)$ lie in the region $Q_l$ (between eigenvalues $\lambda_{l-1}$ and $\lambda_{l+1}$) and below the lower curve of the Dancer-Fučík spectrum. If:
$$\min\{a, b\} > \lambda_l - \frac{S}{|\Omega|^{2s_\sharp/N}}$$
and the "wrong sign" parameter $\gamma$ is sufficiently small, then problem (1.7) admits a nontrivial solution.

Corollaries and Applications:
The authors apply this general theorem to derive new existence results for specific cases:

• Classical Laplacian: Recovers and improves known results for $-\Delta u = bu^+ - au^- + |u|^{2^*-2}u$.
• Fractional Laplacian: New results for $(-\Delta)^s u = \dots$ even in the non-jumping case ($a=b$).
• Mixed Local/Nonlocal: New results for $-\Delta u + (-\Delta)^s u = \dots$.
• Operators with Negative Signs: New results for $-\Delta u - \alpha(-\Delta)^s u = \dots$ (small $\alpha$).
• Infinite Series and Continuous Superposition: Existence for operators defined by convergent series $\sum c_k (-\Delta)^{s_k}$ and continuous integrals $\int f(s)(-\Delta)^s ds$.

5. Significance

This paper represents a significant advancement in the theory of nonlocal elliptic equations.

• Mathematical Novelty: It is the first work to systematically treat critical problems with jumping nonlinearities for general mixed-order operators, including those with signed measures.
• Physical Relevance: The ability to model "wrong sign" operators is particularly relevant for mathematical biology and population dynamics, where different sub-populations may exhibit opposing behaviors (e.g., standard diffusion vs. clustering/inverse diffusion).
• Robustness: The methodology demonstrates that the "bad" behavior of negative coefficients or non-differentiable terms can be controlled quantitatively, opening the door to studying a broader class of competing diffusion models.

In summary, the paper establishes a robust existence theory for a highly general class of critical nonlocal problems, unifying various known results and extending them to previously inaccessible settings involving signed measures and jumping nonlinearities.