Construction of solutions for a critical Grushin problem with competing potentials

We apply the finite-dimentional reduction method to study a critical Grushin problem with competing potentials and we construct infinitely many positive solutions with multi-bubbles for it. Particularly, we resort to the local Pohozaev identities to locate the concentration points of these solutions instead of estimating the derivatives of the reduced functional, since the concentration points can be saddle points. Moreover, we discover that the stable critical point of the potential in nonlinear term plays a more important role than that of the potential in linear term in the process of constructing solutions.