Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type

TL;DR

This paper establishes the existence of infinitely many positive multi-bubbling solutions for a class of critical elliptic Hamiltonian systems in higher dimensions, and proves their non-degeneracy using Pohozaev identities.

Contribution

It constructs unbounded sequences of non-radial solutions and demonstrates their non-degeneracy, filling a gap in the theory of critical elliptic systems in high dimensions.

Findings

Existence of infinitely many positive multi-bubbling solutions.

Solutions can have arbitrarily large energy.

Non-degeneracy of solutions proven using Pohozaev identities.

Abstract

This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -\Delta u_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -\Delta u_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded sequence of non-radial positive vector solutions, whose energy can be made arbitrarily large. Moreover, we prove a type of non-degeneracy result by use of various Pohozaev identities, which is of great interest independently. The indefinite linear operator and strongly coupled nonlinearities make the Hamiltonian-type systems in stark contrast both to the systems of Gradient type and to the single critical elliptic equations in the study of the prescribed curvature problems. It is worth noting that, in higher-dimensional cases $(N\geq5)$, there have been no results on the existence of infinitely many bubbling solutions to critical elliptic systems, either of Hamiltonian or Gradient type.