Lyapunov inequality for an elliptic problem with the Robin boundary condition. (December 2015)

• Record Type: Journal Article
• Authors: Hashizume, Masato; Takahashi, Futoshi
• Is Part Of: Nonlinear analysis. Volume 129(2015)
• Page Start: 189
• Page End: 197
• Publication Date: 2015-12
• Subjects: primary 35J25 -- secondary 35J20 46E35; Lyapunov inequality -- Robin boundary condition -- p-Laplacian -- Elliptic equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.08.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the L q -Lyapunov type inequalities for quasi-linear elliptic problems with the Robin boundary conditions, in which the principal part of the equation is p -Laplacian operator. Similar problems have been considered by several authors for linear elliptic problems with the Dirichlet or Neumann boundary conditions. We show that the critical value of the problem is ( N − 1 ) / ( p − 1 ), where N is the dimension of the domain. We reveal the relation between this critical value and the critical exponent p ∗ of the trace Sobolev embedding. W 1, p ( Ω ) ↪ L p ∗ ( ∂ Ω ) .