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arXiv2026

Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method

In this paper, we investigate the existence and concentration of solutions to a $(p,N)$-Laplace equation in $\mathbb{R}^N$ involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.

math.AP↗

arXiv2025

On $(p,N)$-Laplace multivalued equations with critical exponential nonlinearity in $\mathbb{R}^N$

In this paper, we study the existence of nonnegative solutions for a class of multivalued $(p,N)$-Laplace problems having discontinuous nonlinearity with critical exponential growth in $\mathbb{R}^N$. To demonstrate the existence results, we utilized variational methods for non-differentiable functions.

math.AP↗

arXiv2022

PENTATRON: PErsonalized coNText-Aware Transformer for Retrieval-based cOnversational uNderstanding

Conversational understanding is an integral part of modern intelligent devices. In a large fraction of the global traffic from customers using smart digital assistants, frictions in dialogues may be attributed to incorrect understanding of the entities in a customer's query due to factors including ambiguous mentions, mispronunciation, background noise and faulty on-device signal processing. Such errors are compounded by two common deficiencies from intelligent devices namely, (1) the device not being tailored to individual customers, and (2) the device responses being unaware of the context in the conversation session. Viewing this problem via the lens of retrieval-based search engines, we build and evaluate a scalable entity correction system, PENTATRON. The system leverages a parametric transformer-based language model to learn patterns from in-session customer-device interactions coupled with a non-parametric personalized entity index to compute the correct query, which aids downstream components in reasoning about the best response. In addition to establishing baselines and demonstrating the value of personalized and context-aware systems, we use multitasking to learn the domain of the correct entity. We also investigate the utility of language model prompts. Through extensive experiments, we show a significant upward movement of the key metric (Exact Match) by up to 500.97% (relative to the baseline).

cs.LG↗