Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities

2024/12/14 by Oscar Agudelo, Agudelo, Oscar, Bernhard Ruf +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2412.10812

Abstract

We study the \it Hamiltonian elliptic system \ \beginaligned -Δu amp; = λ|v|^r-1v +|v|^p-1v amp;\hboxin Ω,
-Δv amp; = μ|u|^s-1u +|u|^q-1u amp;\hboxin Ω,
u amp;gt;0, vgt;0 amp;\hboxin Ω,
u amp;=v = 0 amp;\hboxon ∂ Ω, \endaligned . where Ω⊂ \mathbb R^N is a smooth bounded domain, λ and μ are nonnegative parameters and r,s,p,q>0. Our study includes the case in which the nonlinearities in \eqrefHS1-abstract are concave near the origin and convex near infinity, and we focus on the region of non-negative \it pairs of parameters \red(λ,μ) that guarantee exis\-tence and multiplicity of solutions of \eqrefHS1-abstract. \redIn particular, we show the existence of a strictly decreasing curve λ_*(μ) on an interval [0, μ] with λ_*(0)> 0, λ_*(μ) = 0 and such that the system has two solutions for (λ,μ) below the curve, one solution for (λ, μ) on the curve and no solution for (λ, μ) above the curve. A similar statement holds reversing λ and μ. This work is motivated by some of the results by Ambrosseti, BRezis and Cerami from 1993.

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