Towers of Nodal Bubbles for the Bahri–Coron Problem in Punctured Domains

Let Omega be a bounded smooth domain in R-N which contains a ball of radius R centered at the origin, N >= 3. Under suitable symmetry assumptions, for each delta is an element of (0, R), we establish the existence of a sequence (u(m,delta)) of nodal solutions to the critical problem-Delta u = vertical bar u vertical bar(2*-2)u in Omega(delta) := {x is an element of Omega : vertical bar x vertical bar > delta} on partial derivative Omega(delta),where 2* := 2N/N-2 is the critical Sobolev exponent. We show that, if Omega is strictly star-shaped then, for each m is an element of N, the solutions u(m, delta) concentrate and blow up at 0, as delta -> 0, and their limit profile is a tower of nodal bubbles, that is, it is a sum of rescaled nonradial sign-changing solutions to the limit problem-Delta u = vertical bar u vertical bar(2*-2)u , u is an element of D-1,D-2(R-N),centered at the origin.