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We consider the existence of positive bound states for the nonlinear Schrödinger equation −ε 2 Δu + V(x)u = u p , u &gt; 0, u ∈ H 1 (ℝ N ), where N ≥ 3, 1 &lt; p &lt; (N + 2)/(N − 2), and V is a nonnegative potential with compact support. For arbitrary positive integer k ∈ Z+, we construct higher energy solutions with exactly k peaks which interact with each other and cluster around a local maximum point of V when ε is sufficiently small. The main part of the solutions decays exponentially but the perturbation part decays algebraically at infinity.

Bound States with Clustered Peaks for Nonlinear Schrödinger Equations with Compactly Supported Potentials