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...

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} = |x|^a |u|^{p-1} \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending celebrated result of [14], obtain a classification on finite Morse index solutions to elliptic above potential. particular, critical exponent $p$ Joseph–Lundgren type is derived in supercritical ...