Sign-changing solutions for critical Hamiltonian systems in $\mathbb{R}^N$

TL;DR

This paper proves the existence of infinitely many non-radial sign-changing solutions for a class of critical Hamiltonian elliptic systems in rom ocusing on systems with critical exponents on the hyperbola, using novel methods.

Contribution

It introduces new ideas and strategies to establish infinitely many sign-changing solutions for critical Hamiltonian systems without Kelvin invariance.

Findings

Established existence of infinitely many solutions.

Constructed non-radial sign-changing solutions.

Developed new methods applicable to other critical problems.

Abstract

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -\Delta u =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -\Delta v =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the exponents $(p,q)$ satisfy $p,q>1$ and belong to the critical hyperbola $$\frac1{p+1}+\frac1{q+1} =\frac {N-2}N.$$ To establish this result, we introduce several new ideas and strategies that are both robust and potentially applicable to other critical problems lacking the Kelvin invariance.