This work is aimed at studying the plasmon resonances of metallic nanoparticles. We show that these values are the complex eigenvalues of Maxwell's equations that only occur when the dielectric permittivity of the nanoparticles is negative and the size of the nanoparticles d is less than the incident wavelength λ0 , that is δ = d/λ0 << 1 . Afterwards, we prove that the resonances satisfy a nonlinear spectral problem on the boundary of the nanoparticles. Using Fredholm theory and the generalized Rouché Theorem we derive the complete asymptotic of the plasmon resonances as the parameter δ tends to zero.
