Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

1. Problem Statement

The authors investigate a class of quasilinear elliptic equations involving a mixed local-nonlocal operator with variable exponents, a singular term, and a superlinear growth term. The problem is formulated as:

$$\begin{cases} -\Delta_{p(x)}u + (-\Delta)^s_{p(x)} u = \lambda a(x)u^{-\gamma(x)} + b(x)u^{r(x)-1}, & u > 0 \text{ in } \Omega, \\ u = 0, & \text{in } \mathbb{R}^N \setminus \Omega, \end{cases}$$

Key Components:

• Operators:
  • Local: The $p(x)$-Laplacian, $\Delta_{p(x)}u = \text{div}(|\nabla u|^{p(x)-2}\nabla u)$.
  • Nonlocal: The fractional $p(x)$-Laplacian, $(-\Delta)^s_{p(x)} u$, defined via a Cauchy principal value integral with a variable exponent kernel $p(x,y)$.
• Nonlinearities:
  • Singular Term: $\lambda a(x)u^{-\gamma(x)}$ where $0 < \gamma(x) < 1$. This term causes the energy functional to be non-differentiable at zero.
  • Superlinear Term: $b(x)u^{r(x)-1}$ where $p(x) < r(x) < p^*(x)$ (subcritical growth).
• Parameters: $\lambda > 0$ is a parameter; $a(x)$ and $b(x)$ are weight functions.
• Domain: $\Omega \subset \mathbb{R}^N$ is a bounded domain with a Lipschitz boundary.

2. Methodology

The authors employ variational methods combined with topological degree theory and fibering map analysis.

• Functional Framework:

  • The problem is set in a Banach space $X_0(\Omega)$, defined as the intersection of the variable exponent Sobolev space $W^{1,p(x)}_0(\Omega)$ and a fractional Sobolev-type space $X_0$ tailored for the nonlocal operator.
  • The associated energy functional $J_\lambda$ is defined but is not Fréchet differentiable due to the singular term $u^{-\gamma(x)}$.

• Nehari Manifold Approach:

  • Since the functional is not bounded below on the whole space, the authors restrict the search for critical points to the Nehari manifold $\mathcal{N}_\lambda = \{ u \in X_0 \setminus \{0\} : \langle J'_\lambda(u), u \rangle = 0 \}$.
  • They decompose the Nehari manifold into three disjoint sets based on the sign of the second derivative of the fibering map $\phi_u(t) = J_\lambda(tu)$:
    • $\mathcal{N}^+_\lambda$: Local minima of the fibering map ($\phi''_u(1) > 0$).
    • $\mathcal{N}^-_\lambda$: Local maxima of the fibering map ($\phi''_u(1) < 0$).
    • $\mathcal{N}^0_\lambda$: Inflection points ($\phi''_u(1) = 0$).

• Handling Variable Exponents:

  • A significant challenge is the lack of homogeneity due to variable exponents $p(x), r(x), \gamma(x)$.
  • To analyze the monotonicity and existence of critical points for the fibering maps, the authors utilize topological degree theory. They approximate the variable exponent integrals with sums of constant exponent integrals (using Lemma 3.1 and 3.2) to establish strict monotonicity properties required for the fibering map analysis.

• Regularity Analysis:

  • To prove that weak solutions are bounded ($L^\infty$), the authors adapt the De Giorgi iteration method combined with a localization argument. This involves constructing a sequence of level sets and deriving recursive inequalities for the measure of these sets.

3. Key Contributions

1. Novelty in Mixed Operators: This is one of the first studies addressing mixed local-nonlocal operators with variable exponents in the presence of singularities. Previous works mostly focused on constant exponents or purely local/nonlocal cases.
2. Topological Degree Application: The paper introduces a novel technique using topological degree arguments to handle the monotonicity of functionals involving variable exponents, overcoming the difficulty that standard scaling arguments fail in this setting.
3. Decomposition of Nehari Manifold: The authors rigorously prove that for sufficiently small $\lambda$, the set $\mathcal{N}^0_\lambda$ is empty (contains only the zero function), allowing a clean separation of the manifold into $\mathcal{N}^+_\lambda$ and $\mathcal{N}^-_\lambda$.
4. Regularity for Variable Exponents: They establish $L^\infty$ bounds for solutions to mixed local-nonlocal problems with variable exponents, a result not previously available in this specific context.

4. Main Results

Theorem 2.1 (Multiplicity of Solutions):
Under specific assumptions on the exponents ($p(x), r(x), \gamma(x)$) and weight functions, there exists a threshold $\Lambda > 0$ such that for any $\lambda \in (0, \Lambda)$, the problem admits at least two distinct positive weak solutions:

• One solution $u \in \mathcal{N}^+_\lambda$ (associated with a local minimum of the energy).
• One solution $v \in \mathcal{N}^-_\lambda$ (associated with a mountain-pass type geometry).

Theorem 2.2 (Regularity):
If the weight functions satisfy certain integrability conditions (e.g., $a(x) \in L^\infty$ or specific variable exponent Lebesgue spaces), any positive weak solution $u$ belongs to $L^\infty(\Omega)$. This ensures the solutions are essentially bounded.

5. Significance and Impact

• Mathematical Rigor: The paper bridges a gap in the literature by extending the theory of singular elliptic equations to the complex setting of mixed operators with variable growth.
• Physical Modeling: The results are relevant for modeling physical phenomena where material properties vary spatially (variable exponents) and involve both short-range (diffusion) and long-range (Lévy flight) interactions. Examples include image restoration, electrorheological fluids, and biological population dispersal.
• Methodological Advancement: The combination of fibering maps with topological degree theory to handle variable exponents provides a robust framework that can be applied to other non-homogeneous elliptic problems where standard variational techniques face obstacles due to the lack of scaling invariance.

In summary, the paper successfully establishes the existence of multiple solutions for a highly complex class of elliptic equations and provides essential regularity results, advancing the understanding of mixed local-nonlocal problems in variable exponent spaces.