On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem −ɛ2Δu+u=Q(x)|u|q−2u, x∈RN, u∈H1(RN) (1.1) where Δ = ∑Ni=1δ2/δx2i is the Laplace operator in RN, 2 < q < ∞ for N = 1, 2, 2 < q < 2N/(N−2) for N⩾3, and Q(x) is a bounded positive continuous function on RN satisfying the following conditions. (Q1) Q has a strict local minimum at some point x0∈RN, that is, for some δ > 0 Q(x)>Q(x0) for all 0 < ∣x−x0∣ < δ. (Q2) There are constants C, θ > 0 such that |Q(x)−Q(y)|⩽C|x−y|θ for all ∣x−x0∣ ⩽ δ, ∣y−y0∣ ⩽ δ. Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in RN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ε is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as ε → 0 everywhere else in RN.