Acta Mathematica Scientia ( IF 1 ) Pub Date : 2021-06-29 , DOI: 10.1007/s10473-021-0509-2 Pengyan Wang , Pengcheng Niu
In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms
$$\left\{ {\matrix{{( - \Delta )_{{a_1}}^{{\alpha \over 2}}{u_1}(x) = u_1^{{q_{11}}}(x) + u_2^{{q_{12}}}(x) + {h_1}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {( - \Delta )_{{a_2}}^{{\beta \over 2}}{u_2}(x) = u_1^{{q_{21}}}(x) + u_2^{{q_{22}}}(x) + {h_2}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {{u_1}(x) = 0,\,\,\,{u_2}(x) = 0,\,\,\,\,\,\,x \in {\mathbb{R}}{^n}\backslash \Omega .} \hfill \cr } } \right.$$Here \(( - \Delta )_{{a_1}}^{{\alpha \over 2}}\) and \(( - \Delta )_{{a_2}}^{{\beta \over 2}}\) denote weighted fractional Laplacians and Ω ⊂ ℝn is a C2 bounded domain. It is shown that under some assumptions on hi(i = 1, 2), the problem admits at least one positive solution (u1(x), u2(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.
中文翻译：
加权分数系统的先验界限和正解的存在性
在本文中，我们证明了以下加权分数系统的正解的存在性，该系统涉及具有梯度项的不同加权分数拉普拉斯算子
$$\left\{ {\matrix{{( - \Delta )_{{a_1}}^{{\alpha \over 2}}{u_1}(x) = u_1^{{q_{11}}}( x) + u_2^{{q_{12}}}(x) + {h_1}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2 }(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {( - \Delta )_{{a_2}}^{{\beta \over 2} {u_2}(x) = u_1^{{q_{21}}}(x) + u_2^{{q_{22}}}(x) + {h_2}(x,{u_1}(x),{ u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {{ u_1}(x) = 0,\,\,\,{u_2}(x) = 0,\,\,\,\,\,\,x \in {\mathbb{R}}{^n}\反斜杠 \Omega .} \hfill \cr } } \right.$$这里\(( - \Delta )_{{a_1}}^{{\alpha \over 2}}\)和\(( - \Delta )_{{a_2}}^{{\beta \over 2}} \)表示加权分数拉普拉斯算子，Ω ⊂ ℝ n是C 2有界域。结果表明，在对h i ( i = 1, 2) 的一些假设下，该问题至少有一个正解 ( u 1 ( x ), u 2 ( x ))。我们首先通过使用 Chen、Li 和 Li 的直接爆破方法获得系统解的先验界限。那么存在的证明是基于拓扑度理论。
京公网安备 11010802027423号