Construction of solutions for a critical elliptic system of Hamiltonian type

2025/09/14 by Yuxia Guo, Guo, Yuxia, Congzheng Xuanyuan +2
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.11251

openalex publication_date 2025/09/14 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begincases -Δu + V(|y'|,y'') u = |v|^p-1v, amp; in ℝ^N,\newline -Δv + V(|y'|,y'') v = |u|^q-1u, amp; in ℝ^N, \endcases where (y', y'') ∈ ℝ² × ℝ^N-2, V(|y'|, y'') \not≡ 0 is a bounded, nonnegative function on ℝ₊ × ℝ^N-2 and p, q > 1 lie on the critical hyperbola: (1)/(p+1) + (1)/(q+1) = (N-2)/(N). By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that N ≥ 5, (p,q) lies in a certain admissible range, and r² V(r, y'') has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.

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