On the generalized principal eigenvalue of quasilinear operator: definitions and qualitative properties

• Published in: Calculus of variations and partial differential equations 2019-06, Vol.58 (3), p.1-22, Article 102
• Main Authors: Nguyen, Phuoc-Tai, Vo, Hoang-Hung
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Domains; Economic models; Eigenvalues; Linear operators; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Partial differential equations; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-019-1523-2

The notions of generalized principal eigenvalue for linear second order elliptic operators in general domains introduced by Berestycki et al. (Commun Pure Appl Math 47:47–92, 1994 ) and Berestycki and Rossi (J Eur Math Soc (JEMS) 8:195–215, 2006 , Commun Pure Appl Math 68:1014–1065, 2015 ) have become a very useful and important tool in analysis of partial differential equations. This motivates us for our study of various concepts of eigenvalue for quasilinear operator of the form K V [ u ] : = - Δ p u + V u p - 1 , u ≥ 0 . This operator is a natural generalization of self-adjoint linear operators. If Ω is a smooth bounded domain, we already proved in Nguyen and Vo (J Funct Anal 269:3120–3146, 2015) that the generalized principal eigenvalue coincides with the (classical) first eigenvalue of K V . Here we investigate the relation between three types of the generalized principal eigenvalue for K V on general smooth domain (possibly unbounded), which plays an important role in the investigation of their limits with respect to the parameters. We also derive a nice simple condition for the simplicity of the generalized principal eigenvalue and the spectrum of K V in R N . To these aims, we employ new ideas to overcome fundamental difficulties originated from the nonlinearity of p -Laplacian. We also discuss applications of the notions by examining some examples.