A PRIORI BOUNDS AND THE EXISTENCE OF POSITIVE SOLUTIONS FOR WEIGHTED FRACTIONAL SYSTEMS

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doi:10.1007/s10473-021-0509-2

Abstract: In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:

{\begin{cases} (- Δ)_{a_{1}}^{\frac{α}{2}} 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 Ω, \\ (- Δ)_{a_{2}}^{\frac{β}{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 Ω, \\ u_{1} (x) = 0, u_{2} (x) = 0, x \in R^{n} ∖ Ω . \end{cases}

$\left\{\begin{array}{lll} (-\Delta)^{\frac{\alpha}{2}}_{a_1} u_1(x)=u_1^{q_{11}}(x)+u_2^{q_{12}}(x)+ h_1(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ (-\Delta)^{\frac{\beta}{2}}_{a_2} u_2(x)=u_1^{q_{21}}(x)+u_2^{q_{22} }(x)+h_2(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ u_1(x)=0,~u_2(x)=0,~~~x\in \mathbb{R}^n \backslash \Omega. \end{array}\right.$ Here

(- Δ)_{a_{1}}^{\frac{α}{2}}

$(-\Delta)^{\frac{\alpha}{2}}_{a_1}$ and

(- Δ)_{a_{2}}^{\frac{β}{2}}

$(-\Delta)^{\frac{\beta}{2}}_{a_2}$ denote weighted fractional Laplacians and

Ω \subset R^{n}

$\Omega \subset \mathbb{R}^n$ is a

C^{2}

$C^2$ bounded domain. It is shown that under some assumptions on

h_{i} (i = 1, 2)

$h_i(i=1,2)$ , the problem admits at least one positive solution

(u_{1} (x), u_{2} (x))

$(u_1(x),u_2(x))$ . 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.

Key words: weighted fractional system, gradient term, existence, a priori bounds

CLC Number:

35J61

Pengyan WANG, Pengcheng NIU. A PRIORI BOUNDS AND THE EXISTENCE OF POSITIVE SOLUTIONS FOR WEIGHTED FRACTIONAL SYSTEMS[J].Acta mathematica scientia,Series B, 2021, 41(5): 1547-1568.

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