Infinitely many solutions for elliptic system with Hamiltonian type

2025/02/20 by Zhang, Jia, Zhang, Weimin
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.14549

Abstract

In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \begincases \beginaligned -Δuamp;=H_v(u, v) \quadamp;amp;in~Ω,
-Δvamp;=H_u(u, v) \quadamp;amp;in~Ω,
u, vamp;=0~~amp;amp;on ~ ∂Ω,
\endaligned \endcases where N≥ 1, Ω⊂ ℝ^N is a bounded domain and H∈ C¹( ℝ²) is strictly convex, even and subcritical. We mainly present two results: (i) When H is superlinear, the system has infinitely many solutions, whose energies tend to infinity. (ii) When H is sublinear, the system has infinitely many solutions, whose energies are negative and tend to 0. As a byproduct, the Lane-Emden system under subcritical growth has infinitely many solutions.

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