A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

Technical Summary: A Mountain-Pass Algorithm for Nonlocal Problems with Super-Quadratic Nonlinearities

Problem Statement
This paper investigates the existence and numerical approximation of nontrivial solutions to the nonlinear nonlocal equation:
$$-Lu(x) = f(x, u(x))$$
defined on a bounded domain $\Omega \subset \mathbb{R}^n$. The operator $L$ is a nonlocal integral operator generated by a radially symmetric $L^1$ convolution kernel $\gamma$:
$$Lu(x) = \int_{\mathbb{R}^n} (u(y) - u(x))\gamma(|x-y|) \, dy.$$
The study considers both homogeneous nonlocal Dirichlet ($u=0$ on $\Omega^c$) and Neumann ($Lu=0$ on $\Omega^c$) boundary conditions. The nonlinearity $f$ is assumed to be super-quadratic, satisfying specific growth and structural conditions that distinguish it from standard sub-critical or linear cases. The authors aim to address a gap in the literature where minimax theory has been applied to fractional Laplacians but not extensively to general nonlocal operators with $L^1$ kernels that may possess sign-changing properties or algebraic decay.

Methodology and Theoretical Framework
The paper employs a variational approach, reformulating the problem as finding critical points of an associated energy functional $I: X \to \mathbb{R}$:
$$I[u] = \frac{1}{2}B[u, u] - \int_{\Omega} F(x, u) \, dx,$$
where $B$ is a bilinear form derived from the operator $L$, and $F$ is the primitive of $f$.

1. Operator Decomposition and Functional Spaces:
   The authors assume the kernel $\gamma$ can be decomposed into a diffusive part $\gamma_e$ and a perturbation $\gamma_s$. The diffusive part generates an operator whose Fourier symbol behaves like $-|\xi|^2$ near the origin (mimicking the Laplacian) and satisfies analyticity conditions. This decomposition allows the authors to define the problem on the Sobolev space $H^1(\Omega)$ rather than $L^2(\Omega)$. This is a crucial technical step, as $L^2$ lacks the necessary compactness for the Mountain Pass Theorem in this setting. By establishing that the bilinear forms associated with Dirichlet and Neumann conditions are bounded and coercive on $H^1(\Omega)$, the authors ensure the validity of the variational framework.

2. Existence via Mountain Pass Theorem:
   The primary theoretical tool is the classical Mountain Pass Theorem. The authors verify the necessary geometric conditions:

   • Geometry (G1 & G2): The energy functional is positive on a sphere of small radius and negative at some point far from the origin. This relies on the super-quadratic growth of $f$ (specifically Assumption A4) and a modified scaling condition (Assumption A3) that replaces the standard Ambrosetti-Rabinowitz condition, allowing for non-odd nonlinearities.
   • Compactness (Palais-Smale Condition): The authors prove that any sequence $\{u_n\}$ with bounded energy and vanishing derivative possesses a convergent subsequence. This is achieved by leveraging the compact embedding of $H^1(\Omega)$ into $L^p(\Omega)$ and the specific growth bounds on $f$.

3. Numerical Scheme:
   To compute solutions, the authors adapt a gradient descent algorithm tailored to the "mountain pass" geometry. Unlike standard Newton methods, which may fail to find nontrivial solutions or converge to trivial ones in this context, the proposed algorithm exploits the energy landscape:

   • Radial Maximization: For a given direction $u$, the algorithm explicitly computes the scalar $t^*$ that maximizes the energy along the ray $tu$.
   • Angular Descent: The algorithm then updates the direction by solving a linearized nonlocal equation to find a descent direction in the "angular" component.
   • Iteration: This process repeats until the gradient norm falls below a tolerance.

Key Results

• Existence Theorem: The paper proves that under mild assumptions on the kernel (including sign-changing and algebraically decaying kernels) and super-quadratic nonlinearities, the nonlocal problem admits at least one nontrivial weak solution in $H^1(\Omega)$ for both Dirichlet and Neumann boundary conditions.
• Numerical Validation: The authors present numerical simulations using finite element methods for various kernels (exponentially decaying, algebraically decaying, and inverted Mexican Hat) and nonlinearities (including $f(u)=u^3$, $u^5$, and $u^3-u$).
  • The algorithm demonstrates convergence in $L^1$ and $L^2$ norms of the residual and error.
  • The method successfully finds nontrivial solutions where standard Newton iterations fail or produce large residuals.
  • The scheme handles sign-changing kernels (relevant for neural field models) and non-odd nonlinearities effectively.

Significance and Claims
The authors claim this work is the first to apply minimax theory (specifically the Mountain Pass Theorem) to nonlocal problems defined by general $L^1$ convolution kernels with finite second moments, rather than restricting the analysis to fractional Laplacians. A significant contribution is the reformulation of the operator to generate bilinear forms on $H^1(\Omega)$, thereby securing the compactness required for critical point theory in a nonlocal setting.

Furthermore, the paper highlights the practical necessity of the proposed algorithm. The authors note that standard Newton-type solvers often fail to approximate true nontrivial solutions for these specific nonlocal equations, frequently converging to trivial solutions or exhibiting poor residual behavior. The proposed gradient-descent-based scheme, which leverages the specific geometry of the energy functional, provides a robust alternative for finding these solutions.

The paper concludes by noting that while the current work establishes existence and a numerical method, future work could explore similar multiplicity results for different geometric structures (e.g., Linking Theorem) and extend the framework to optimal control problems with nonlocal constraints.