论文标题
多项式生长组的非线性choquard方程的溶液的存在和收敛
The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth
论文作者
论文摘要
在本文中，我们研究了非线性choquard方程\开始{eqnarray*}δ^{2} u-Δu+（1+λa（x）） dimension $ n \ geq 2 $，其中$α\ in（0，n），\，\，p> \ frac {n+α} {n} {n}，\，λ$是一个正参数，$r_α$代表着绿色的离散分数laplacian的函数，该功能与Riesz的潜在相同。根据$ a（x）$的某些假设，我们通过nehari歧管方法建立了基态解决方案对非线性choquard方程的存在和渐近行为。
In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} Δ^{2}u-Δu+(1+λa(x))u=(R_α\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N\geq 2$, where $α\in(0,N),\,p>\frac{N+α}{N},\,λ$ is a positive parameter and $R_α$ stands for the Green's function of the discrete fractional Laplacian, which has same asymptotics as the Riesz potential. Under some assumptions on $a(x)$, we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.
