A Priori Bounds and the Existence of Positive Solutions for Weighted Fractional Systems

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

$$\left\{ {\matrix{{( - \Delta )_{{a_1}}^{{\alpha \over 2}}{u_1}(x) = u_1^{{q_{11}}}(x) + u_2^{{q_{12}}}(x) + {h_1}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {( - \Delta )_{{a_2}}^{{\beta \over 2}}{u_2}(x) = u_1^{{q_{21}}}(x) + u_2^{{q_{22}}}(x) + {h_2}(x,{u_1}(x),{u_2}(x),\nabla {u_1}(x),\nabla {u_2}(x)),\,\,\,\,\,\,x \in \Omega ,} \hfill \cr {{u_1}(x) = 0,\,\,\,{u_2}(x) = 0,\,\,\,\,\,\,x \in {\mathbb{R}}{^n}\backslash \Omega .} \hfill \cr } } \right.$$

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 h_i(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.