Multiple interior peak solutions for some singularly perturbed neumann problems

We consider the problem

$$\{\begin{array}{c}{\epsilon }^2\Delta u-u+f(u)=0,u>0,in\Omega ,\\ \frac{\partial u}{\partial v}=0on\partial \end{array}$$

, where Ω is a bounded smooth domain in R^N, ɛ > 0 is a small parameter, and ƒ is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as ɛ approaches zero, at a critical point of the mean curvature function H(P), P ε ∂Ω. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function d(P, ∂Ω), P ε Ω. In this paper, we prove the existence of interior K −peak (K ⩾ 2) solutions at the local maximum points of the following function φ(P₁, P₂,…, p_k) = min_{i, k, l = 1, …, K; k ≠ 1} (d(P_i, ∂Ω), 1/2 ¦P_k −p_l¦). We first use the Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function φ(P₁, …, p_k) appears naturally in the asymptotic expansion of the energy functional.