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A PRIORI BOUNDS AND THE EXISTENCE OF POSITIVE SOLUTIONS FOR WEIGHTED FRACTIONAL SYSTEMS
Pengyan WANG, Pengcheng NIU
Acta mathematica scientia,Series B. 2021, 41 (5):
1547-1568.
DOI: 10.1007/s10473-021-0509-2
In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:$$\left\{\begin{array}{lll} (-\Delta)^{\frac{\alpha}{2}}_{a_1} u_1(x)=u_1^{q_{11}}(x)+u_2^{q_{12}}(x)+ h_1(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ (-\Delta)^{\frac{\beta}{2}}_{a_2} u_2(x)=u_1^{q_{21}}(x)+u_2^{q_{22} }(x)+h_2(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ u_1(x)=0,~u_2(x)=0,~~~x\in \mathbb{R}^n \backslash \Omega. \end{array}\right.$$ Here $(-\Delta)^{\frac{\alpha}{2}}_{a_1}$ and $(-\Delta)^{\frac{\beta}{2}}_{a_2}$ denote weighted fractional Laplacians and $\Omega \subset \mathbb{R}^n$ is a $C^2$ bounded domain. It is shown that under some assumptions on $h_i(i=1,2)$, the problem admits at least one positive solution $(u_1(x),u_2(x))$. We first obtain the {a priori} bounds of solutions to the system by using the direct blow-up method of Chen, Li and Li. Then the proof of existence is based on a topological degree theory.
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