In this paper, we first establish decay estimates for the fractional and higher-order H\'enon-Lane-Emden systems by using a nonlocal average integral estimates, which deduce result of non-existence. Next, apply method scaling spheres introduced in \cite{DQ2} to derive Liouville type theorem. We also construct an interesting example on super $\frac{\alpha}{2}$-harmonic functions (Proposition 1.2).

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value...

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in ODE case, with some new perspectives both theoretical and applied side. We make full use invariance up to a divergence, or, as call it here, Bessel-Hagen (BH) invariance. By recognizing that function need not be total time derivative, can easily deduce nonlocal motion. prove always trivial...

By using critical point theory, some new sufficient conditions for the existence of at least 2N distinct solutions to boundary value problems of a discrete generalized Emden–Fowler equation are obtained. © 2009 Elsevier Ltd. All rights reserved.