ON POSITIVE MULTIPEAK SOLUTIONS OF A NONLINEAR ELLIPTIC PROBLEM

Abstract

In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem

formula here

where Δ = [sum ]^N_i=1δ²/δx²_i is the Laplace operator in ℝ^N, 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 ℝ^N satisfying the following conditions.

(Q₁) Q has a strict local minimum at some point x₀∈ℝ^N, that is, for some δ > 0

formula here

for all 0 < [mid ]x−x₀[mid ] < δ.

(Q₂) There are constants C, θ > 0 such that

formula here

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 ℝ^N, 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 ℝ^N.