In this paper we consider the quasilinear Schr\"{o}dinger equation\[-\Delta u+V(x)u-\Delta(u^{2})u=g(x,u), \quad x\in \mathbb{R}^{N},\] where $g$ and $V$are periodic in $x_1,\ldots,x_N$ and $g$ is odd in $u$, subcritical and satisfies a monotonicity condition. We employ the approach developed in [15,16] and obtain infinitely many geometrically distinct solutions.

A complete analysis of the essential spectrum of matrix-differential operators A of the form (-d/dt p d/dt + q -d/dt b* + c*) b d/dt + c D) in L2((alpha, beta)) circle times (L-2((alpha, beta)))(n) singular at beta is an element of R boolean OR {infinity} is given; the coefficient functions p, q are scalar real-valued with p > 0, b, c are vector-valued, and D is Hermitian matrix-valued. The so-called singular part of the essential spectrum sigma(s)(ess)(A) is investigated systematically. Our main results include an explicit description of sigma(s)(ess)(A), criteria for its absence and presence; an analysis of its topological structure and of the essential spectral radius. Our key tools are: the asymptotics of the leading coefficient pi(center dot, lambda) = p - b* (D - lambda)(-1) b of the first Schur complement of (0.1), a scalar differential operator but non-linear in lambda; the Nevanlinna behaviour in lambda of certain limits t NE arrow beta of functions formed out of the coefficients in (0.1). The efficacy of our results is demonstrated by several applications; in particular, we prove a conjecture on the essential spectrum of some symmetric stellar equilibrium models.

Based on a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip, we obtain a uniform upper bound for the number of zeros of solutions to an ordinary differential equation near its Fuchsian singular point, provided that any two distinct characteristic exponents at this point have distinct real parts. The latter result implies that a Fuchsian differential equation with polynomial coefficients is globally non-oscillating in CP1 if and only if every its singular point satisfies the above condition.