Existence and local uniqueness of bubbling solutions for the Grushin critical problem.
The authors consider the Grushin critical problem \[-\Delta u(y,z)=\Phi(y,z)\frac{u^{\frac{N}{N-2}}(y,z)}{|y|},\quad u>0,\quad(y,z)\in\mathbb{R}^k\times\mathbb{R}^h\] where \(k-1\geq h\geq 1\), \(k+h=N\geq 5\), and \(\Phi\in C^1(\mathbb{R}^N)\) is a nonnegative bounded nonconstant function, which is \(1\)-periodic in its the \(\bar{k}\) variables \(z_1,\dots,z_{\bar{k}}\), where \(1\leq\bar{k}<\frac{N-2}{N}\), and satisfies the following conditions: \(0\) is a critical point of \(\Phi\), with \(\Phi(0)>0\), and there exist \(\beta\in(N-2,N-1)\), \(\theta>0\), \(a_1,\dots,a_N\in\mathbb{R}\setminus\{0\}\), with \(\sum_{i=1}^Na_i<0\), and \(R\in C^{1,[\beta]}(\mathbb{R}^N)\) satisfying \(\sum_{s=0}^{[\beta]}|\nabla^sR(x)||x|^{-\beta+s}=O(|x|^{\theta})\) as \(x\rightarrow 0\), such that \[\Phi(x)=\Phi(0)+\sum_{i=1}^Na_i|x_i|^{\beta}+R(x),\quad\text{for all }|x|\text{ small}.\] With these assumptions, the authors show that, if \(\{P_i\}_{i\in\mathbb{N}}\) is a sequence of points in \(\mathbb{R}^{\bar{k}}\times\{0_{\mathbb{R}^{N-k-\bar{k}}}\}\) satisfying certain properties, then the problem admits a solution \(u\) with infinitely many bubbles concentrating at \(P_1,P_2,P_3,\dots\). To prove the existence of such a solution \(u\), the authors first consider the case of a finite sequence of points \(P_1,\dots,P_n\) and find a solution \(u_n\) with bubbles concentrating at \(P_1,P_2,\dots,P_n\). Then, using elliptic estimates, they get the solution \(u\) as a limit in \(C^2_{loc}(\mathbb{R}^N)\) of the sequence \(\{u_n\}\). Next, the authors prove a local uniqueness property which, in turn, they use to derive the periodicity with respect to the variables \(z_1,\dots,z_{\bar{k}}\) of the bubbling solution \(u\).
scientific article; zbMATH DE number 7031709
scientific article