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

Phuong Le

doi:10.3934/dcds.2020333

Article Contents

2021, Volume 41, Issue 4: 1605-1626. Doi: 10.3934/dcds.2020333

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

Phuong Le

Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Received: April 2020

Revised: August 2020

Early access: September 2020

Published: April 2021

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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Abstract

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.

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

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