We consider the following singularly perturbed Neumann problem: ε2 Δu - u + f(u) = 0 in Ω, ∂u/∂u = 0 on ∂Ω, where Δ = Σi=1N ∂2/∂i2 is the Laplace operator, ε > 0 is a constant, Ω is a bounded, smooth domain in ℝN with its unit outward normal v, and f is superlinear and subcritical. A typical f is f (u) = up 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/εN(|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/εN(|ln ε|N] number of solutions. Moreover, for each m ε (0, N) there exist solutions with energies in the order εN-m.
ASJC Scopus subject areas
- Applied Mathematics