An index classification theory of homogeneous p-Laplacian equations and existence of solutions of non-homogeneous equations

In this paper, we first investigate the classification of positively homogeneous equations (phi p(u '))' + q(t)phi p(u) = 0, u(0) = 0 = u(1), where p > 1. is fixed, phi p(u) = |u|(p-2)u and q epsilon L-infinity (0, 1), and then discuss the existence of solutions for non-homogeneous equations. The train method of classification is by using a generalized Prufer equation theta ' = |cos(p)theta|(p) + q(t)/p-1 sin(p) theta|(p) for t epsilon (0, 1), where sin(p) : R -> [-1, 1] is a periodic function and cos(p) t = d sin(p) t/dt for t epsilon R.