Interior Peak Solutions for a Semilinear Dirichlet Problem

In this paper, we consider the semilinear Dirichlet problem (P-epsilon):-Delta u+V(x)u=u(n+2/)n-2-epsilon, u>0 in Omega, u=0 on partial derivative Omega, where Omega is a bounded regular domain in Rn, n >= 4, epsilon is a small positive parameter, and V is a non-constant positive C2-function on Omega<overline>. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for (P epsilon). The proof of our results relies on precise expansions of the gradient of the Euler-Lagrange functional associated with (P epsilon), along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems.