The spectrum for the p-Laplacian systems with rotating periodic boundary value conditions and applications

In this paper, we study the spectrum problem for the p-Laplacian system (phi p(u'))' + lambda phi p(u) = 0 with rotating periodic boundary value conditions u(T) = Qu(0), u'(T) = Qu ' ( 0 ) , where Q is an N x N orthogonal real matrix and 0 represents the zero vector. We give the definitions of spectrum and eigenvalue for the above problem, prove the relationship between them, discuss the basic properties of the eigenvalues and obtain the variational characteristic of the first positive eigenvalue, which deduce the generalized Wirtinger inequality and Sobolev inequality. These results provide a theoretical basis for subsequent research. In order to use the variational method, we establish a Sobolev space corresponding to rotating periodic boundary value conditions and give some useful properties. As applications of the spectrum of the above problem, we prove the existence of solutions for the p-Laplacian system rotating periodic boundary value problem with growth of order p and the rotating periodic Li & eacute;nard type system involving p-Laplacian. Our results extend and enrich the existing relative work. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.