Construction of solutions to a nonlinear critical elliptic system via local Pohozaev identities - Guo, Qidong; Wang, Qingfang; Wu, Wenju (Topological Methods in Nonlinear Analysis, 2025)
In this paper, we investigate the following elliptic system with Sobolev critical growth\begin{equation*} \begin{cases} -\Delta u+P(|y'|,y'')u=u^{2^*-1}+\dfrac{\beta}{2}u^{{2^*}/{2}-1}v^{{2^*}/{2}}, & y\in \mathbb{R}^N,\vspace{0.13cm}\\ -\Delta v+Q(|y'|,y'')v=v^{2^*-1}+\dfrac{\beta}{2} v^{{2^*}/{2}-1}u^{{2^*}/{2}}, & y\in \mathbb{R}^N,\vspace{0.13cm}\\ u, v> 0, & u,v\in H^1\left(\mathbb{R}^N\right), \end{cases}\end{equation*}where $(y',y'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$, $P(|y'|,y'')$,$ Q(|y'|,y'')$ are boundednon-negative function in $\mathbb{R}^+\times\mathbb{R}^{N-2}$, $2^*={2N}/({N-2})$.By combining a finite reduction argument and local Pohozaev type of identities, assuming that $N\geq 5$ and $r^2(P(r,y'')+\kappa ^2Q(r,y''))$ has a common topologically nontrivial critical point,we construct an unbounded sequence of non-radial positive vector solutions of synchronized type, whose energy can be made arbitrarily large. Our result extends the result of a single critical problem by Chen, Pistoia and Vaira(Discrete Contin. Dyn. Syst. {\bf 43} (2023), 482-506).The novelties mainly include the following two aspects.On one hand, when $N\geq5$, the coupling exponent ${2}/({N-2})< 1$, which creates a great trouble for us to apply the perturbation argument directly.This constitutes the main difficulty different between the coupling system anda single equation.On the other hand, the weaker symmetry conditions of $P(y)$ and $Q(y)$ make us not estimate directly the corresponding derivatives of the reducedfunctional in locating the concentration points of the solutions, we employ some local Pohozaevidentities to locate them.