Interior Peak Solutions for a Semilinear Dirichlet Problem

In this paper, we consider the semilinear Dirichlet problem (Pε):−Δu+V(x)u=un+2n−2−ε, u>0 in Ω, u=0 on ∂Ω, where Ω is a bounded regular domain in Rn, n≥4, ε is a small positive parameter, and V is a non-constant positive C2-function on Ω¯. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for (Pε). The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with (Pε), 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.