In this paper we continue our investigation in [5, 7, 8] on multipeak
solutions to the problem
where Δ =
is the Laplace operator in ℝN, 2 < q < ∞ for
N = 1, 2, 2 < q < 2N/(N−2) for
N[ges ]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
for all 0 < [mid ]x−x0[mid ] < δ.
(Q2) There are constants C, θ > 0 such that
for all [mid ]x−x0[mid ] [les ] δ,
[mid ]y−y0[mid ] [les ] 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