A multiplicity theorem for a class of nonlocal elliptic equations involving even nonlinearities

Technical Summary: A Multiplicity Theorem for Nonlocal Elliptic Equations with Even Nonlinearities

Problem Statement
The paper investigates a Dirichlet boundary value problem for a class of nonlocal elliptic equations on a bounded smooth domain $\Omega \subset \mathbb{R}^n$. The problem, denoted as $(P)$, is formulated as:
$$\begin{cases} -\Delta u = Q\left(\int_{\Omega} F(x, u(x))dx\right) f(x, u) + g(x, u) & \text{in } \Omega \\ u = 0 & \text{on } \partial\Omega \end{cases}$$
Here, $F(x, t) = \int_0^t f(x, s)ds$, and the nonlocal term arises from the function $Q$ depending on the integral of $F$ over the domain. The study focuses on the existence of weak solutions in the Sobolev space $H^1_0(\Omega)$. The primary objective is to establish conditions under which the problem admits at least three distinct weak solutions for parameters $\lambda$ running within a suitable open interval.

Methodology and Functional Framework
The author employs variational methods, specifically utilizing critical point theory. The weak solutions of $(P)$ are identified as critical points of the energy functional $\Phi: H^1_0(\Omega) \to \mathbb{R}$, defined as:
$$\Phi(u) = \frac{1}{2}\|u\|^2 - J(u)$$
where $\|u\|$ is the standard norm in $H^1_0(\Omega)$, and $J(u)$ incorporates the nonlocal term and the perturbation $g$.

The core of the proof relies on a strict minimax inequality and a three critical point theorem (referenced as Theorem 3 in [2]). The methodology proceeds through the following steps:

1. Construction of an Auxiliary Functional: The author introduces a functional $P(u, \lambda)$ involving a strictly convex function $\alpha$, a continuous odd function $\gamma$, and the nonlocal integral term.
2. Verification of the Minimax Inequality: The central technical hurdle is proving the strict inequality:
   $$\sup_{\lambda \in \mathbb{R}} \inf_{u \in H^1_0(\Omega)} P(u, \lambda) < \inf_{u \in H^1_0(\Omega)} \sup_{\lambda \in \mathbb{R}} P(u, \lambda)$$
   This is achieved by contradiction. The author assumes equality and demonstrates that the resulting saddle point $(u^*, \lambda^*)$ leads to a logical inconsistency. Specifically, the proof utilizes the symmetry properties of the functions (evenness of $f$, oddness of $\gamma$) and the strict convexity of $\alpha$ to show that if a solution exists under the equality assumption, it must be trivial ($u^*=0$), which contradicts the construction of the functional for large parameters.
3. Coercivity and Continuity: The proof establishes that the functional $u \mapsto \sup_{\lambda} P(u, \lambda)$ is sequentially weakly lower semicontinuous and coercive. This ensures the existence of minimizers and allows the application of the abstract critical point theorem.
4. Perturbation by $g$: The final step involves adding a perturbation term $g \in \mathcal{A}$ (a class of Carathéodory functions with specific growth conditions). The author shows that if the oscillation of the primitive of $g$ is sufficiently small (bounded by a constant $\epsilon^*_\mu$), the strict minimax inequality is preserved, guaranteeing the multiplicity of solutions.

Key Assumptions
The main result (Theorem 1) requires the following conditions:

• Nonlinearity $f$: $f(x, \cdot)$ must be even for almost every $x$, and the set where $f$ is non-zero must have positive measure. Furthermore, $F(x, t)$ must satisfy a sub-quadratic growth condition at infinity ($\lim_{|t|\to\infty} \sup |F(x,t)|/t^2 = 0$).
• Nonlinearity $\gamma$: $\gamma$ is continuous, odd, vanishes only at 0, and its primitive satisfies a linear growth bound.
• Convex Function $\alpha$: $\alpha$ is strictly convex, derivable, with $\alpha(0)>0$, $\alpha'(0)=0$, and $\alpha'(R)=R$.
• Parameter $\mu$: The parameter $\mu$ must be sufficiently large.
• Perturbation $g$: The difference between the integral of the supremum and the integral of the infimum of the primitive of $g$ must be less than a specific threshold $\epsilon^*_\mu$.

Main Results
The paper proves Theorem 1, which states that under the aforementioned assumptions, there exists a threshold $\epsilon^*_\mu > 0$ such that for any perturbation $g$ satisfying the smallness condition, there exists an open interval $A \subseteq \mathbb{R}$ and a radius $\rho > 0$. For every $\lambda \in A$, the problem $(P)$ possesses at least three weak solutions in $H^1_0(\Omega)$, all with norms strictly less than $\rho$.

Significance and Claims
The paper positions itself as a "very short note" aiming to prove a specific multiplicity result for nonlocal problems involving even nonlinearities.

• Novelty: The author explicitly states in Remark 1 that they are "not aware of known results close enough to Theorem 1 so that a proper comparison can be made." This suggests the result fills a gap in the literature regarding the specific combination of nonlocal terms, even nonlinearities, and the three-solution multiplicity via strict minimax inequalities.
• Scope: The work is purely theoretical, focusing on the existence of solutions within the variational framework. It does not propose numerical methods, specific physical applications, or future experimental directions. The significance lies in the extension of critical point theory to a specific class of nonlocal elliptic equations where the nonlinearity exhibits even symmetry.