We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.

AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

The use of Raman spectroscopy as a characterization tool for individual single wall carbon nanotubes is briefly reviewed. New physical phenomena occurring at the single nanotube level are discussed, with special emphasis given to the use of resonance Raman scattering for the structural determination of (n, m) for individual nanotubes, based on diameter and chirality dependent phenomena associated with the radial breathing mode, the G-band and the G[prime]-band features. Examples are given to show how single nanotube spectroscopy provides insight into the use of Raman spectroscopy for the characterization of nanotube bundles and for the study of new physical phenomena occurring at the single nanotube level.