Existence, Local uniqueness and periodicity of bubbling solutions for a critical nonlinear elliptic equation

TL;DR

This paper investigates the existence, uniqueness, and periodicity of bubbling solutions for a critical nonlinear elliptic equation with a periodic potential, revealing conditions under which solutions are symmetric, periodic, or non-existent.

Contribution

It establishes the existence and local uniqueness of infinitely many bubbling solutions with periodicity properties for the elliptic equation with a periodic potential.

Findings

Existence of infinitely many bubbling solutions under certain conditions.

Local uniqueness implies symmetry and periodicity of solutions.

Non-existence results when the potential's minimum is not zero.

Abstract

We revisit the following nonlinear critical elliptic equation \begin{equation*} -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the periodicity of bubbling solutions. Here we try to investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose blow-up set is $\{(jL,0,...,0):j=0,\pm 1, \pm 2,..., \pm m\}$ must be periodic in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the bubbles which is large enough but independent of $L.$ Moreover, we also show a non-existence of this bubbling solutions for the problem above if the local minimum of $Q(y)$ does not equal to zero.