We consider the eigenvalue problem for the following p -Laplacian-like equation: − div ( a ( | D u | p ) | D u | p − 2 D u ) = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω ⊂ ℝ n is a bounded smooth domain. When λ is small enough, a multiplicity result for eigenfunctions are obtained. Two examples from nonlinear quantized mechanics and capillary phenomena, respectively, are given for applications of the theorems.