In this work we consider the existence of positive singular solutions style='text-indent:20px;'> \begin{document}$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ u_2 \lambda_2 u_1|^q 0 on \partial \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $\...

We classify solutions of finite Morse index of the fractional LaneEmden equation (−∆)su = |u|p−1u in R.

We present a useful analytic approximation to the solution of the Lane-Emden equation for infinite polytropic index the isothermal sphere. The optimized expression obtained for the density profile can be accurate to within 0.04% within 5 core-radii and to 0.1% within 10 core-radii.

In this work we propose a novel approach to integrate the Lane-Emden equations for relativistic anisotropic polytropes. We take advantage of fact that Gravitational Decoupling allows decrease number degrees freedom once well known solution Einstein field is provided as seed so after demanding polytropic equation radial pressure system automatically closed. The not only extend both isotropic or ...

In this paper we study the existence of uniform a priori estimates for positive solutions to Navier problems higher order Lane–Emden equations \begin{equation}\label{0-0} (-\Delta)^{m}u(x)=u^{p}(x), \quad x\in\Omega, \end{equation} all large exponents $p$, where $\Omega\subset\mathbb{R}^{n}$ is star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$, and $2\leq m\leq\fr...

We have derived exact solutions of the isothermal Lane–Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are relevant to the dynamics of the solar nebula before and during the stages of planet and satellite formation. The nonrotating solution for the mass density is analytic, nonsingular, monotonically decreasing with radius, and it sat...

Introduction The investigation of Lane Emden HLEEL equation has a long standing history Hsee, e.g., @1 10D, to mention only part of a vast literatureL. Recently Schaudt @6D has shown that Picard type iteration scheme may be used to show the existence, uniqueness and regularity of global solutions of LEE , in particular, for n 3 1. In this paper it was shown that straightforward usage of Picard ...