Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions

1. Problem Statement

The paper investigates the existence of nontrivial weak solutions for a nonlinear elliptic equation involving a linear operator of mixed order, subject to homogeneous Neumann boundary conditions.

The Operator:
The core object of study is the operator $L_{\alpha, \mu}$, defined as a superposition of the classical Laplacian and a continuous (or discrete) superposition of fractional Laplacians:
$$L_{\alpha, \mu}(u) := -\alpha\Delta u + \int_{(0,1)} (-\Delta)^s u \, d\mu(s)$$
where:

• $\alpha \geq 0$ is a coefficient for the classical Laplacian.
• $\mu$ is a nonnegative, finite Borel measure over $(0, 1)$.
• $(-\Delta)^s$ denotes the fractional Laplacian of order $s$.
• The framework allows for discrete sums (e.g., $\sum c_k (-\Delta)^{s_k}$), continuous integrals, or combinations thereof.

The Equation:
The authors study the following problem in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$:
$$\begin{cases} L_{\alpha, \mu}(u) + u = \lambda u + f(x, u) & \text{in } \Omega \\ \text{Homogeneous } (\alpha, \mu)\text{-Neumann conditions} & \text{on } \partial\Omega \text{ and } \mathbb{R}^N \setminus \Omega \end{cases}$$
Here, $\lambda \in \mathbb{R}$ is a parameter, and $f(x, u)$ is a nonlinear source term satisfying subcritical growth conditions.

Boundary Conditions:
The Neumann conditions are nonlocal. For $x \in \mathbb{R}^N \setminus \Omega$, the condition involves the integral of the nonlocal normal derivatives:
$$\int_{(0,1)} \mathcal{N}_s u(x) \, d\mu(s) = 0$$
If $\alpha > 0$, the classical local Neumann condition $\partial_\nu u = 0$ on $\partial\Omega$ is also imposed.

2. Methodology and Functional Setting

Functional Spaces:
To handle the mixed-order operator and nonlocal boundary conditions, the authors introduce a specialized functional setting $H_{\alpha, \mu}(\Omega)$.

• The space is defined based on the Gagliardo seminorms integrated against the measure $\mu$.
• Crucial Distinction: For the case $\lambda \geq \lambda_1$ (where $\lambda_1$ is the first eigenvalue), the authors introduce a subspace $\tilde{H}_{\alpha, \mu}(\Omega)$. This subspace is defined to ensure the existence of a complete orthogonal system of eigenfunctions, which is necessary for the Linking Theorem.
  • Specifically, $\tilde{H}_{\alpha, \mu}(\Omega)$ consists of functions in the main space that satisfy the homogeneous nonlocal boundary condition in a strong sense (vanishing of the integrated nonlocal normal derivative).
  • The paper proves that this subspace is closed with respect to weak convergence, a non-trivial result given the nonlocal nature of the boundary conditions.

Variational Framework:
The problem is treated variationally. Solutions correspond to critical points of the energy functional $I: H_{\alpha, \mu}(\Omega) \to \mathbb{R}$:
$$I(u) = \frac{1}{2}\|u\|_{\alpha, \mu}^2 - \frac{\lambda}{2}\|u\|_{L^2(\Omega)}^2 - \int_\Omega F(x, u) \, dx$$
where $F(x, t) = \int_0^t f(x, \tau) d\tau$.

Nonlinearity Assumptions:
The function $f$ satisfies:

1. Subcritical Growth: $|f(x, t)| \leq a_1 + a_2|t|^{p-1}$ with $p < 2^*_{s_\sharp}$, where $s_\sharp$ is a "critical exponent" determined by the measure $\mu$ (specifically, the supremum of the support of $\mu$ if $\alpha=0$, or $1$ if $\alpha > 0$).
2. Ambrosetti-Rabinowitz (AR) Condition: Ensures superlinear growth at infinity to guarantee the boundedness of Palais-Smale sequences.
3. Additional Growth Condition: A lower bound on the primitive $F$ to handle cases where $\lambda$ is large.

Eigenvalue Analysis:
The existence theory is bifurcated based on the position of the parameter $\lambda$ relative to the spectrum of the operator $L_{\alpha, \mu} + \text{Id}$.

• Let $0 = \tilde{\lambda}_1 < \tilde{\lambda}_2 \leq \dots$ be the eigenvalues of $L_{\alpha, \mu}$.
• The relevant eigenvalues for the shifted operator are $\lambda_k = \tilde{\lambda}_k + 1$. Note that $\lambda_1 = 1$.

3. Key Contributions and Results

The paper establishes existence theorems for two distinct regimes of the parameter $\lambda$, utilizing different topological methods.

Case 1: $\lambda < \lambda_1$ (Mountain Pass Geometry)

• Method: The Mountain Pass Theorem.
• Logic:
  • The functional $I$ satisfies the geometry of a mountain pass: it has a local minimum at $u=0$ and a direction where the energy goes to $-\infty$.
  • The Palais-Smale (PS) condition is verified. The authors prove that any sequence with bounded energy and vanishing gradient is bounded in the functional space, and subsequently converges strongly.
  • This relies on the fact that for $\lambda < 1$, the quadratic part of the functional is coercive.
• Result (Theorem 1.2): For any $\lambda < \lambda_1$, there exists a nontrivial weak solution in $H_{\alpha, \mu}(\Omega)$.

Case 2: $\lambda \geq \lambda_1$ (Linking Geometry)

• Method: The Linking Theorem.
• Challenge: When $\lambda \geq \lambda_1$, the functional is no longer coercive in the direction of the first eigenfunction. The standard space $H_{\alpha, \mu}(\Omega)$ does not guarantee a complete orthogonal system of eigenfunctions needed for the linking structure.
• Innovation: The authors restrict the problem to the subspace $\tilde{H}_{\alpha, \mu}(\Omega)$. They prove that this subspace admits a complete orthogonal system of eigenfunctions $\{e_k\}$, allowing for a direct sum decomposition:
  $$\tilde{H}_{\alpha, \mu}(\Omega) = H_i \oplus H_i^\perp$$
  where $H_i = \text{span}\{e_1, \dots, e_i\}$.
• Logic:
  • The functional is bounded from below on the infinite-dimensional subspace $H_i^\perp$ (due to $\lambda < \lambda_{i+1}$).
  • The functional is non-positive on a finite-dimensional manifold linking $H_i$ and a direction in $H_i^\perp$.
  • The PS condition is verified in this subspace.
• Result (Theorem 1.3): For any $\lambda \geq \lambda_1$, there exists a nontrivial weak solution in $\tilde{H}_{\alpha, \mu}(\Omega)$.

4. Applications and Examples (Section 5)

The generality of the measure $\mu$ allows the main theorems to recover and extend several specific cases previously unstudied or treated separately:

1. Mixed Local-Nonlocal Operators: $\alpha \Delta + \beta (-\Delta)^s$ (Dirac measure).
2. Finite Sums of Fractional Operators: $\sum_{k=1}^n (-\Delta)^{s_k}$.
3. Infinite Series of Operators: $\sum_{k=0}^\infty c_k (-\Delta)^{s_k}$.
4. Continuous Superpositions: $\int_0^1 \omega(s)(-\Delta)^s ds$.

In all these cases, the paper provides the first existence results for Neumann boundary conditions with subcritical nonlinearities, extending previous work that was often limited to Dirichlet conditions or critical exponents.

5. Significance and Novelty

• Unified Framework: The paper provides a single, rigorous framework to handle operators of any mixed order (discrete or continuous) under Neumann conditions, a setting that was previously fragmented in the literature.
• New Functional Analysis: The introduction of the subspace $\tilde{H}_{\alpha, \mu}(\Omega)$ and the proof of its weak closedness and completeness of eigenfunctions are significant technical contributions. This resolves the difficulty of applying linking arguments to nonlocal operators with Neumann conditions.
• Boundary Condition Handling: The work rigorously handles the "nonlocal normal derivative" for superpositions of operators, generalizing definitions from previous works (e.g., [DPLV23], [DPLSV24]).
• Extension of Criticality: The paper carefully defines the "critical exponent" $2^*_{s_\sharp}$ for mixed operators, showing how the highest order in the superposition dictates the growth limits of the nonlinearity.

Conclusion:
This paper successfully bridges the gap between classical elliptic theory and modern nonlocal analysis for complex, mixed-order operators. By adapting variational methods (Mountain Pass and Linking) to a newly constructed functional setting, it establishes the existence of solutions for a broad class of nonlinear problems that were previously out of reach, particularly in the context of Neumann boundary conditions.