An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $\Delta_2$--condition

1. Problem Statement

The paper investigates the existence and asymptotic behavior of eigenvalues for a class of nonlocal, quasilinear, anisotropic operators defined in the context of fractional Orlicz-Sobolev spaces.

The primary operator studied is $L_a^s$, defined for a function $u$ as:
$$L_a^s u(x) = 2 \text{p.v.} \int_{\mathbb{R}^n} a\left(x, y, \frac{u(x) - u(y)}{|x-y|^s}\right) \frac{dy}{|x-y|^{n+s}}$$
where $s \in (0, 1)$ is the fractional order, and $a(x, y, \xi)$ is a kernel function satisfying specific structural conditions (oddness, symmetry, monotonicity, and growth).

The associated eigenvalue problem is:
$$\begin{cases} L_a^s u = \lambda g(u) & \text{in } \Omega \\ u = 0 & \text{in } \mathbb{R}^n \setminus \Omega \end{cases}$$
where $\Omega \subset \mathbb{R}^n$ is a bounded open set, $\lambda$ is the eigenvalue, and $g$ is a nonlinear source term.

The Core Challenge:
The study is conducted in Fractional Orlicz-Sobolev spaces ($W^{s, M}_0(\Omega)$) generated by a Young function $M$. Crucially, the authors do not assume the $\Delta_2$-condition on the Young function $M$ (or its conjugate $\bar{M}$).

• Implication: Without the $\Delta_2$-condition, the Orlicz space $L_M$ is generally non-reflexive and non-separable. Consequently, the standard variational methods (like the classical Ljusternik–Schnirelmann theory) and the differentiability of the associated energy functionals fail, as the dual space structure is more complex.

2. Methodology

To overcome the lack of reflexivity and separability, the authors employ a sophisticated combination of functional analysis and variational techniques:

1. Complementary Systems:
   The authors utilize the theory of complementary systems (introduced by Gossez and developed by others). They construct a system $(W^{s, M}_0, W^{s, E}_0; W^{-s, \bar{M}}, W^{-s, \bar{E}})$ where $E_M$ is the closure of bounded functions in $L_M$. This allows them to work with the separable subspace $W^{s, E}_0$ while retaining the duality structure necessary for analysis in the larger non-reflexive space.

2. Pseudomonotonicity of the Operator:
   A central technical achievement is proving that the operator $L_a^s$ is pseudomonotone with respect to the weak topology $\sigma(W^{s, M}_0, W^{-s, \bar{E}})$.

   • This property is essential for passing to the limit in nonlinear terms when the space is not reflexive.
   • The proof relies on the monotonicity of the kernel $a$, growth conditions, and a specific "segment property" of the domain $\Omega$.

3. Galerkin Approximation:
   Since the full space is not separable, the authors cannot apply standard critical point theory directly. Instead, they use a Galerkin-type approximation:

   • They restrict the problem to a sequence of finite-dimensional subspaces $V_k$ spanned by a countable dense subset of $W^{s, E}_0$.
   • On these finite-dimensional spaces, the standard Ljusternik–Schnirelmann theory applies, yielding a sequence of eigenpairs $(\lambda_{k}^i, u_{k}^i)$.

4. Limit Passage:
   Using the pseudomonotonicity of $L_a^s$ and the compactness properties of the embedding $W^{s, M}_0(\Omega) \hookrightarrow E_M(\Omega)$, they prove that the sequence of approximate eigenpairs converges (up to a subsequence) to a true eigenpair in the infinite-dimensional space.

3. Key Contributions

• Generalization of Nonlocal Operators: The paper generalizes previous results on the fractional $p$-Laplacian and fractional $m$-Laplacian to a fully anisotropic and non-standard growth setting defined by a general function $a(x, y, \xi)$.
• Removal of the $\Delta_2$-Condition: This is the most significant theoretical contribution. The authors successfully establish the existence of an infinite sequence of eigenvalues without requiring the Young function $M$ to satisfy the $\Delta_2$-condition. This extends the applicability of eigenvalue theory to spaces with very slow or very fast growth (e.g., exponential growth) where standard reflexivity fails.
• Pseudomonotonicity in Non-Reflexive Spaces: The proof of the pseudomonotonicity of $L_a^s$ in the specific topology $\sigma(W^{s, M}_0, W^{-s, \bar{E}})$ provides a robust framework for solving nonlocal problems in non-reflexive Orlicz spaces.
• Asymptotic Behavior: The paper rigorously characterizes the behavior of the eigenvalues and eigenfunctions as the index $k \to \infty$.

4. Main Results

Theorem 4.10 (Main Result):
Let $\Omega$ be a bounded open domain satisfying the segment property. Then, there exists a sequence of eigenpairs $\{(\lambda_i, u_i)\}_{i=1}^\infty$ for the problem (1.4) such that:

1. Existence: Each $(\lambda_i, u_i)$ is a solution in the sense of distributions, with $u_i \in \text{Dom}(L_a^s) \subset W^{s, M}_0(\Omega)$.
2. Eigenvalue Asymptotics: The eigenvalues tend to infinity:
   $$\lambda_i \to +\infty \quad \text{as } i \to \infty$$
3. Eigenfunction Asymptotics: The eigenfunctions converge to zero in the weak topology:
   $$u_i \to 0 \quad \text{in } \sigma(W^{s, M}_0(\Omega), W^{-s, \bar{E}}(\Omega)) \quad \text{as } i \to \infty$$

5. Significance and Impact

• Mathematical Rigor: The paper bridges a gap in the theory of nonlocal PDEs by handling the "pathological" case where the underlying function space lacks reflexivity. This is a non-trivial extension of the classical spectral theory for the Laplacian and $p$-Laplacian.
• Physical Modeling: The operators studied model non-standard growth diffusion processes. Many physical phenomena (e.g., in plasticity, fluid dynamics, or image processing) exhibit growth rates that do not satisfy the $\Delta_2$-condition. This work provides the theoretical foundation for analyzing eigenmodes in such complex media.
• Methodological Framework: The combination of complementary systems, Galerkin approximations, and pseudomonotonicity arguments presented here serves as a template for analyzing other nonlinear, nonlocal problems in non-reflexive Banach spaces.

In summary, the paper successfully establishes the spectral theory for a broad class of nonlocal quasilinear operators in the most general fractional Orlicz setting, overcoming the major technical hurdle of the missing $\Delta_2$-condition.