Construction of solutions for the polyharmonic equation via local Pohozaev identities

We consider the following problem involving critical exponent and polyharmonic operator: (-Delta)(m)u + V(vertical bar y'vertical bar, y '')u = u(m*-1), u > 0, u is an element of D-m,D-2 (R-N), where m* = 2N/N-2m, N >= 4m + 1, m is an element of N+, (y', y '') is an element of R-2 x RN-2 and V(vertical bar y'vertical bar , vertical bar y ''vertical bar) is a bounded non-negative function in R+ x RN-2. By using a finite reduction argument and local Pohozaev type identities, we will show that if N >= 4m + 1 and r(2m)V(r, y '') has a stable critical point (r(0), y(0)''), then the above problem has infinitely many solutions, whose energy can be arbitrarily large.