Maximum and comparison principles to Lane‐Emden systems

Abstract

This paper focuses on maximum and comparison principles related to the Lane–Emden problem {−scriptL1u=λρfalse(xfalse)|v|α−1vinΩ,−scriptL2v=μτfalse(xfalse)|u|β−1uinΩ,u=v=0on∂Ω,where α,β>0, αβ=1, normalΩ is a smooth bounded open subset of Rn with n⩾1, ρ and τ are positive functions on normalΩ and L1 and L2 are second‐order uniformly elliptic linear operators in normalΩ. We characterize the couples false(λ,μfalse)∈double-struckR2 such that the weak maximum principle associated to this problem holds in normalΩ. Moreover, weak and strong versions of maximum and comparison principles are also proved. As applications, we establish existence and uniqueness of solution for the non‐homogeneous counterpart of the above problem as well as Aleksandrov–Bakelman–Pucci‐type estimates. Lower bounds for principal eigenvalues of Lane–Emden systems in terms of the measure of normalΩ are also derived.

On Hamiltonian systems with critical Sobolev exponents

“…Item c) deserves some extra comments, since the most accurate 1-homogenous perturbation to Hamiltonian systems, given below in (HS), is induced by the hyperbola of points (r, s) such that rs = 1. Indeed, this hyperbola has been named as the spectral curve for Hamiltonian systems; see (MONTENEGRO, 2000;LEITE;MONTENEGRO, 2019;LEITE;MONTENE-GRO, 2020) for linear operators and (SANTOS et al, 2020) in the fully nonlinear scenario. In this thesis we address these three questions and present some results observed in the framework of Hamiltonian systems which are non-existent for scalar problems.…”