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Construction of solutions for a critical elliptic system of Hamiltonian type

Yuxia Guo, Congzheng Xuanyuan, Tingfeng Yuan

TL;DR

This work tackles the existence of infinitely many bubbling solutions for a nonlinear elliptic Hamiltonian system with critical growth on $\mathbb{R}^N$. It develops a Lyapunov–Schmidt finite-dimensional reduction combined with local Pohozaev identities and Green's representation to control bubble locations and corrections. Under $N\ge 5$, $p,q>1$ on the critical hyperbola and a nondegenerate critical point of $r^2V(r,y'')$, the authors construct bubbling solutions with energy that can be made arbitrarily large, with the bubble count $m$ serving as a parameter. The approach circumvents compactness issues by locating bubbles via local Pohozaev constraints, leading to an infinite family of positive solutions concentrating near a stable critical point of the potential, and it extends multiplicity results to coupled Hamiltonian systems with critical exponents. The results broaden the understanding of critical elliptic systems and provide a framework for analyzing energy blow-up and concentration phenomena in noncompact settings.

Abstract

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ is a bounded, nonnegative function on $\mathbb{R}_+ \times \mathbb{R}^{N-2}$ and $p, q > 1$ lie on the critical hyperbola: \[ \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] 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 $N \ge 5$, $(p,q)$ lies in a certain admissible range, and $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.

Construction of solutions for a critical elliptic system of Hamiltonian type

TL;DR

This work tackles the existence of infinitely many bubbling solutions for a nonlinear elliptic Hamiltonian system with critical growth on . It develops a Lyapunov–Schmidt finite-dimensional reduction combined with local Pohozaev identities and Green's representation to control bubble locations and corrections. Under , on the critical hyperbola and a nondegenerate critical point of , the authors construct bubbling solutions with energy that can be made arbitrarily large, with the bubble count serving as a parameter. The approach circumvents compactness issues by locating bubbles via local Pohozaev constraints, leading to an infinite family of positive solutions concentrating near a stable critical point of the potential, and it extends multiplicity results to coupled Hamiltonian systems with critical exponents. The results broaden the understanding of critical elliptic systems and provide a framework for analyzing energy blow-up and concentration phenomena in noncompact settings.

Abstract

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where , is a bounded, nonnegative function on and lie on the critical hyperbola: 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 , lies in a certain admissible range, and has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.

Paper Structure

This paper contains 7 sections, 28 theorems, 355 equations.

Table of Contents

  1. Introduction
  2. Finite Dimensional Reduction
  3. Proof of Main Result
  4. Asymptotic behavior and non-degeneracy of Ground State
  5. Estimate of the approximate solution
  6. The energy expansion
  7. Some technical Estimates

Key Result

Theorem 1.1

Suppose that $V \in C^1(\mathbb{R}^N)$ is bounded, nonnegative, and satisfies condition (V). If $N\geq 6$, $p\in(\frac{N}{N-2},\frac{N+2}{N-2})$ or $N=5$, $p\in (2,\frac{7}{3})$, and $p, q$ satisfy critical hyperbola, then problem Main admits infinitely many solutions whose energy can be made arbitr

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • ...and 36 more