论文标题
多项式生长组的choquard类型方程的阳性基态解决方案存在
The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth
论文作者
论文摘要
在本文中，让$ g $是一组离散的多项式增长的Cayley图，具有同质尺寸$ n \ geq3 $。我们在$ g $：\ begin {equation}ΔU+（r_α\ ast \ ast u \ mid u \ mid^{p} {p}）\ mid u \ mid u \ mid u \ mid u \ mid u \ mid u \ re \ n = 0，\ end eend {equation}中，我们研究choquard类型方程代表绿色的离散分段拉普拉斯操作员的功能，该功能与Riesz电位具有相同的渐近剂。我们证明了此类cayley图上的离散耐铁木 - - 贝布莱夫的不平等，并且通过离散的浓度 - 乳胶原理，我们证明了在超临界情况下相应的Sobolev型不平等的极端函数，从而产生上述choquard类型方程的正状态解决方案。此外，我们通过$ p $ laplace，biharmonic和$ p $ -biharmonic运营商等获得了Choquard类型方程的积极基态解决方案。
In this paper, let $G$ be a Cayley graph of a discrete group of polynomial growth with homogeneous dimension $N\geq3$. We study the Choquard type equation on $G$: \begin{equation} Δu+(R_α\ast\mid u\mid^{p})\mid u\mid^{p-2}u=0, \end{equation} where $α\in(0,N)$, $p>\frac{N+α}{N-2}$ and $R_α$ stands for the Green's function of the discrete fractional Laplace operator, which has same asymptotics as the Riesz potential. We prove the discrete Hardy-Littlewood-Sobolev inequality on such Cayley graphs, and by the discrete Concentration-Compactness principle we prove the existence of extremal functions for the corresponding Sobolev type inequalities in supercritical cases, which yields a positive ground state solution of the above Choquard type equation. Moreover, we obtain positive ground state solutions of Choquard type equations with $p$-Laplace, biharmonic and $p$-biharmonic operators etc.
