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A capacity-based condition for existence of solutions to fractional elliptic equations with first-order terms and measures | | María Laura de Borbón
; Pablo Ochoa
; | | Date: |
13 Apr 2020 | | Abstract: | In this manuscript, we appeal to Potential Theory to provide a sufficient
condition for existence of distributional solutions to fractional elliptic
problems with non-linear first-order terms and measure data $omega$: $$
left{ egin{array}{rcll} (-Delta)^su&=&|
abla u|^q + omega quad ext{in
}mathbb{R}^n,, ,,s in (1/2, 1)\u & > &0 quad ext{in }
mathbb{R}^{n}\lim_{|x| o infty}u(x) & =& 0, end{array}
ight. $$under
suitable assumptions on $q$ and $omega$. Roughly speaking, the condition for
exis-tence states that if the measure data is locally controlled by the Riesz
fractional capacity, then there is a global solution for the equation. We also
show that if a positive solution exists, necessarily the measure $omega$ will
be absolutely continuous with respect to the associated Riesz capacity, which
gives a partial reciprocal of the main result of this work. Finally, estimates
of $u$ in terms of $omega$ are also given in different function spaces. | | Source: | arXiv, 2004.6057 | | Services: | Forum | Review | PDF | Favorites | |
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