Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation−Δu+V(|y′|,y″)u=uN+2N−2,u>0,u∈H1(RN), where (y′,y″)∈R2×RN−2, V(|y′|,y″) is a bounded non-negative function in R+×RN−2. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if N≥5 and r2V(r,y″) has a stable critical point (r0,y0″) with r0>0 and V(r0,y0″)>0, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.