New type of solutions to a slightly subcritical Henon type problem on general domains

Abstract

We consider the following slightly subcritical problem

((sic)epsilon) { -Delta u = beta(x)vertical bar u vertical bar(p-1-epsilon) u in Omega, u = 0 on partial derivative Omega,

where Omega is a smooth bounded domain in R-n, 3 <= n <= 6, p := n+2/n-2 is the Sobolev critical exponent, epsilon is a small positive parameter and beta is an element of C-2 ((Omega) over bar) is a positive function. We assume that there exists a nondegenerate critical point xi(*) is an element of partial derivative Omega of the restriction of p to the boundary partial derivative Omega such that

del(beta(xi(*)) -2/p-1) . eta(xi(*)) > 0,

where eta denotes the inner normal unit vector on partial derivative Omega. Given any integer k >= 1, we show that fors epsilon > 0 small enough problem ((sic)epsilon) has a positive solution, which is a sum of k bubbles which accumulate at xi(*) as epsilon tends to zero. We also prove the existence of a sign changing solution whose shape resembles a sum of a positive bubble and a negative bubble near the point xi(*.)