On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded

Article type: Research Article

Authors: Esposito, Luca^a | Roy, Prosenjit^{b; *} | Sk, Firoj^b

Affiliations: [a] DipMat, University of Salerno, Italy. E-mail: luesposi@unisa.it | [b] Indian Institute of Technology, Kanpur, India. E-mails: prosenjit@iitk.ac.in, firoj@iitk.ac.in

Correspondence: [*] Corresponding author. E-mail: prosenjit@iitk.ac.in.

Abstract: We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.

Keywords: Eigenvalues problems, ℓ goes to plus infinity, dimension reduction

DOI: 10.3233/ASY-201626

Journal: Asymptotic Analysis, vol. 123, no. 1-2, pp. 79-94, 2021