Solutions for nonhomogeneous fractional ( <i>p</i> , <i>q</i> )-Laplacian systems with critical nonlinearities
Mengfei Tao, Binlin Zhang
Abstract
Abstract In this article, we aimed to study a class of nonhomogeneous fractional ( p , q )-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional ( p , q )-Laplacian systems in the case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>=</m:mo> <m:mi>s</m:mi> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mi>l</m:mi> <m:mi>q</m:mi> </m:math> N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional ( p , q )-Laplacian systems is the main novelty of this article.