Differential and Integral Equations

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

Billel Gheraibia, Chunhua Wang, and Jing Yang

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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

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

First available in Project Euclid: 11 December 2018

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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