Nonlinear Neumann eigenvalues in outward cuspidal domains with weighted measure

TL;DR

This paper investigates the nonlinear Neumann eigenvalue problem in outward cuspidal domains with weighted measures, establishing embeddings and solvability results that lead to estimates of eigenvalues.

Contribution

It introduces new Sobolev space embeddings into weighted Lebesgue spaces, enabling the analysis of the spectral problem in complex cuspidal domains.

Findings

Embeddings of Sobolev spaces into weighted Lebesgue spaces established

Solvability of the nonlinear Neumann spectral problem demonstrated

Weighted Neumann eigenvalue estimates derived

Abstract

We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These embeddings give the solvability of the Neumann spectral problem in this setting and provide estimates for the corresponding weighted Neumann eigenvalues.