Multi-bump solutions for a semilinear Schroedinger equation

Abstract

We study the existence of multi-bump solutions for the semilinear Schrodinger equation -Δu + (1 + ea(x))u = |u| p-2 u, u ∈ H 1 (ℝ N ), where N ≥ 1, 2 2 if N = 1 or N = 2, and e > 0 is a parameter. The function a is assumed to satisfy the following conditions: a ∈ C(ℝ N ), a(x) > 0 in ℝ N , a(x) = 0(1) and ln(a(x)) = 0(|x|) as lxl → ∞. For any positive integer n, we prove that there exists e(n ) > 0 such that, for 0 < e < e(n), the equation has an n-bump positive solution. Therefore, the equation has more and more multi-bump positive solutions as e → 0.