Early and reliable detection of Alzheimer's disease (AD) is crucial for timely clinical intervention and improved patient management. It also supports the evaluation of emerging therapeutic strategies. In this paper, we propose a Low-Rank Mixture of Experts (LoRA-MoE) deep learning framework for Alzheimer's disease diagnosis based on handwriting analysis. Handwriting signals provide a non-invasive and scalable digital biomarker that captures subtle cognitive-motor impairments associated with early AD progression. The proposed architecture allows multiple experts to specialize in different handwriting patterns while sharing a common base network. This design enables efficient learning of general representations while reducing interference between experts. Each expert is equipped with lightweight low-rank adapters. This mechanism significantly reduces the number of trainable parameters compared with standard Mixture of Experts (MoE) models and improves training stability. The proposed framework is evaluated on the Diagnosis AlzheimeR WIth haNdwriting (DARWIN) dataset. Extensive experiments are conducted, including ablation studies on key architectural parameters such as hidden dimension size, number of experts, and LoRA rank. The method is compared with multilayer perceptron (MLP) and conventional MoE architectures. In addition, stacking ensemble strategies (StackMean and StackMax) are investigated to improve robustness and predictive performance. Experimental results show that the LoRA-MoE framework achieves powerful diagnostic performance while activating significantly fewer parameters during inference. These results highlight the potential of the proposed approach as an accurate and computationally efficient solution for handwriting-based Alzheimer's disease screening and digital health applications.