## Differential and Integral Equations

### Existence and local uniqueness of bubbling solutions for the Grushin critical problem

#### Abstract

In this paper, we study the following Grushin critical problem $$-\Delta u(x)=\Phi(x)\frac{u^{\frac{N}{N-2}}(x)} {|y|},\,\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb R^{N},$$ where $x=(y,z)\in\mathbb R^{k}\times \mathbb R^{N-k},N\geq 5,\Phi(x)$ is positive and periodic in its the $\bar{k}$ variables $(z_{1},...,z_{\bar{k}}),1\leq \bar{k} < \frac{N-2}{2}.$ Under some suitable conditions on $\Phi(x)$ near its critical point, we prove that the problem above has solutions with infinitely many bubbles. Moreover, we also show that the bubbling solutions obtained in our existence result are locally unique. Our result implies that some bubbling solutions preserve the symmetry from the potential $\Phi(x).$

#### Article information

Source
Differential Integral Equations, Volume 32, Number 1/2 (2019), 49-90.

Dates
First available in Project Euclid: 11 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.die/1544497286

Mathematical Reviews number (MathSciNet)
MR3909979

Zentralblatt MATH identifier
07031709

#### Citation

Gheraibia, Billel; Wang, Chunhua; Yang, Jing. Existence and local uniqueness of bubbling solutions for the Grushin critical problem. Differential Integral Equations 32 (2019), no. 1/2, 49--90. https://projecteuclid.org/euclid.die/1544497286