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This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation , with the general hypotheses on the nonlinearity fbeing as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

We consider the existence of positive bound states for the nonlinear Schrödinger equation −ε2Δu + V(x)u = up, u > 0, u ∈ H1(ℝN), where N ≥ 3, 1 < p < (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.