## Differential and Integral Equations

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

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

Billel Gheraibia, Chunhua Wang, and Jing Yang

#### 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

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35B45: A priori estimates 35J40: Boundary value problems for higher-order elliptic equations

#### 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