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Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.22
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  • We classify all nonnegative nontrivial classical solutions to the equation

    $ (-\Delta)^{\frac{\alpha}{2}} u = c_1 \left(\frac{1}{|x|^{n-\beta}} * f(u)\right) g(u) + c_2 h(u) \quad\text{ in } \mathbb{R}^n, $

    where $ 0<\alpha,\beta<n $, $ c_1,c_2\ge0 $, $ c_1+c_2>0 $ and $ f,g,h \in C([0, +\infty),[0, +\infty)) $ are increasing functions such that $ {f(t)}/{t^{\frac{n+\beta}{n-\alpha}}} $, $ {g(t)}/{t^{\frac{\alpha+\beta}{n-\alpha}}} $, $ {h(t)}/{t^{\frac{n+\alpha}{n-\alpha}}} $ are nonincreasing in $ (0, +\infty) $. We also derive a Liouville type theorem for the equation in the case $ \alpha\ge n $. The main tool we use is the method of moving spheres in integral forms.

    Mathematics Subject Classification: 35R11, 35J30, 35B06, 35B53, 35A02.


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