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تسجيل مستخدم جديدOn the fractional Lane-Emden equation
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We classify solutions of finite Morse index of the fractional Lane- Emden equation
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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={mat
hbb 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 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 fini
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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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