A system of local/nonlocal p-Laplacians: The eigenvalue problem and its asymptotic limit as p→∞

Article type: Research Article

Authors: Buccheri, S.^{a; *} | da Silva, J.V.^b | de Miranda, L.H.^c

Affiliations: [a] Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria | [b] Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, UNICAMP, Cidade Universitária Zeferino Vaz, 13083-859, Campinas, SP, Brazil | [c] Departamento de Matemática, Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, 70910-900, Brasília, DF, Brazil

Correspondence: [*] Corresponding author. E-mail: stefano.buccheri@univie.ac.at.

Abstract: In this work, given p∈(1,∞), we prove the existence and simplicity of the first eigenvalue λp and its corresponding eigenvector (up,vp), for the following local/nonlocal PDE system (0.1)−Δpu+(−Δ)pru=2αα+βλ|u|α−2|v|βuin Ω−Δpv+(−Δ)psv=2βα+βλ|u|α|v|β−2vin Ωu=0on RN∖Ωv=0on RN∖Ω, where Ω⊂RN is a bounded open domain, 0<r,s<1 and α(p)+β(p)=p. Moreover, we address the asymptotic limit as p→∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ∞, and the uniformly convergence of the pair (up,vp) to the ∞-eigenvector (u∞,v∞). Finally, the triple (u∞,v∞,λ∞) verifies, in the viscosity sense, a limiting PDE system.

Keywords: First eigenvalue problem, simplicity, local/nonlocal p-Laplacians, Hölder ∞-Laplacian and ∞-Laplacian

DOI: 10.3233/ASY-211702

Journal: Asymptotic Analysis, vol. 128, no. 2, pp. 149-181, 2022