Differential and Integral Equations

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

Billel Gheraibia, Chunhua Wang, and Jing Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation