Multi-piece of bubble solutions for a nonlinear critical elliptic equation

We revisit the following nonlinear critical elliptic equation -Delta u +V (vertical bar y'vertical bar,y '')u =u(N+2 /N-2), u > 0, u is an element of H-1(R-N), where (y', y '') is an element of R-3 x RN-3, V (vertical bar y vertical bar, y '') is a bounded non-negative function in R+ x RN-3. Assuming that r(2)V (r, y '')has a stable critical point (r(0), y(0)'') with r(0) > 0 and V (r(0), y(0)'') > 0, by using a modified finitedimensional reduction method and various local Pohozaev identities, we prove that the problem above has multi-piece of bubble solutions, whose energy can be made arbitrarily large. Since there involves a new variable (h) over bar in the concentrated points {x(j)(+/-)}(j=1)(m) during the reduction process, we have obtain a more precise estimate for the error term. And the bubble solutions are centered at points lying on the top and the bottom circles of a cylinder. Particularly, in one of these cases, the bubble solutions can concentrate at a pair of symmetric points relative to the origin. Our results present a new clustering type of blow-up phenomenon and we think the reason why this phenomenon can occur is mainly because that the function (r)2V (r, y '') has non-isolated critical points. (c) 2024 Elsevier Inc. All rights reserved.