On the number of interior peak solutions for a singularly perturbed neumann problem

We consider the following singularly perturbed Neumann problem: ε² Δu - u + f(u) = 0 in Ω, ∂u/∂u = 0 on ∂Ω, where Δ = Σ_i=1^N ∂²/∂_i² is the Laplace operator, ε > 0 is a constant, Ω is a bounded, smooth domain in ℝ^N with its unit outward normal v, and f is superlinear and subcritical. A typical f is f (u) = u^p where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N - 2) when N ≥ 3. We show that there exists an ε₀ > 0 such that for 0 < ε < ε₀ and for each integer K bounded by 1 ≤ K ≤ ^αN,Ω,f/ε^N(|ln ε|^N where ^αN,Ω,f is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for ^αN,Ω,f is also given.) As a consequence, we obtain that for ε sufficiently small, there exists at least [^αN, Ω,f/ε^N(|ln ε|^N] number of solutions. Moreover, for each m ε (0, N) there exist solutions with energies in the order ε^N-m.