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Register a new userA new critical curve for the Lane-Emden system
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We study stable positive radially symmetric solutions for the Lane-Emden system $-Delta u=v^p$ in $R^N$, $-Delta v=u^q$ in $R^N$, where $p,qgeq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
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We classify solutions of finite Morse index of the fractional Lane- Emden equation
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
We study positive solutions to the fractional Lane-Emden system begin{equation*} tag{S}label{S} left{ begin{aligned} (-Delta)^s u &= v^p+mu quad &&text{in } Omega (-Delta)^s v &= u^q+ u quad &&text{in } Omega u = v &= 0 quad &&text{in } Omega^c={mathbb R}^N setminus Omega, end{aligned} right. end{equation*} where $Omega$ is a $C^2$ bounded domains in ${mathbb R}^N$, $sin(0,1)$, $N>2s$, $p>0$, $q>0$ and $mu,, u$ are positive measures in $Omega$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $mu$ and $ u$. Furthermore, if $p,q in (1,frac{N+s}{N-s})$ and $0 leq mu,, uin L^r(Omega)$ for some $r>frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.
We have derived exact solutions of the isothermal Lane--Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are most relevant to the dynamics of the protosolar 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 satisfies easily the usual physical boundary conditions at the center. When differential rotation is added to the Lane--Emden equation, an entire class of exact solutions for the mass density appears. We have determined all of these solutions analytically as well. Within this class, solutions that are power laws or combinations of power laws are not capable of satisfying the associated boundary--value problem, but they are nonetheless of profound importance because they constitute baselines to which the actual solutions approach when the central boundary conditions are imposed. Numerical integrations that enforce such physical boundary conditions show that the actual radial equilibrium density profiles emerge from the center close to the nonrotating solution, but once they cross below the corresponding baselines, they cease to be monotonic. The actual solutions are forced to oscillate permanently about the baseline solutions without ever settling onto them because the central boundary conditions strictly prohibit such settling, even in the asymptotic regime of large radii. Based on our results, we expect that quasistatically--evolving protoplanetary disks should develop oscillatory density profiles in their midplanes during their isothermal phase. The peaks in these profiles correspond to local potential minima and their locations are ideal sites for the formation of protoplanets ...
We propose and study a one-dimensional $2times 2$ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.
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