Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities

This paper deals with the following prescribed scalar curvature problem − Δ u = Q ( | y ′ | , y ″ ) u N + 2 N − 2 , u > 0 , y = ( y ′ , y ″ ) ∈ R 2 × R N − 2 , where Q ( y ) is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if N ≥ 5 and Q ( r , y ″ ) has a stable critical point ( r 0 , y 0 ″ ) with r 0 > 0 and Q ( r 0 , y 0 ″ ) > 0 , then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bump solutions. Moreover, the concentration points of the bump solutions include a saddle point of Q ( y ) .