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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={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 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 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.
This paper deals with a nonlinear degenerate parabolic equation of order $alpha$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $alpha=4$ while the Porous Medium Equation is the limit $alpha=2$. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.
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