Construction of solutions to a nonlinear critical elliptic system via local Pohozaev identities

Abstract

In this paper, we investigate the following elliptic system with Sobolev critical growth $-\Delta u+P(|y'|,y'')u=u^{2^*-1}+\frac{\beta}{2} u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}},\ y\in R^N$, $-\Delta v+Q(|y'|,y'')v=v^{2^*-1}+\frac{\beta}{2} v^{\frac{2^*}{2}-1}u^{\frac{2^*}{2}}$, $y\in R^N ,u,v>0,u,\,v\in H^1(R^N), $ where~$(y',y'')\in R^2 \times R^{N-2}$, $P(|y'|,y''), Q(|y'|,y'')$ are bounded non-negative function in $R^+\times R^{N-2}$, $2^*=\frac{2N}{N-2}$. By combining a finite reduction argument and local Pohozaev type of identities, assuming that $N\geq 5$ and $r^2(P(r,y'')+\kappa ^2Q(r,y''))$ have 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 [Peng, Wang and Yan,J. Funct. Anal. 274: 2606-2633, 2018]. The novelties mainly include the following two aspects. On one hand, when $N\geq5$, the coupling exponent $\frac{2}{N-2}<1$, which creates a great trouble for us to apply the perturbation argument directly. This constitutes the main difficulty different between the coupling system and a 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 reduced functional in locating the concentration points of the solutions, we employ some local Pohozaev identities to locate them.