论文标题
存在用于$ \ mathbb r^n $上本地非线性的准椭圆系统的解决方案
Existence of solutions for a quasilinear elliptic system with local nonlinearity on $\mathbb R^N$
论文作者
论文摘要
在本文中，我们研究了一类准线性椭圆系统的解决方案的存在 \ begin {eqnarray*} \ begin {case} {ccc} - \ mbox {div}（ϕ_1（| \ nabla u |）\ nabla u）+v_1（x）ϕ_1（| u |） - \ mbox {div}（ϕ_2（| \ nabla v |）\ nabla v）+v_2（x）ϕ_2（| v |）v =λf_v（x，x，u，u，v），\ \ \ x \ in \ mathbb r^n， u \ in W^{1，φ_1}（\ Mathbb r^n），v \ in W^{1，φ_2}（\ Mathbb r^n）， \ end {cases} \ end {eqnarray*}其中$ n \ ge 2 $，$ \ inf _ {\ mathbb r^n} v_i（x）> 0，i = 1,2 $，和$λ> 0 $。我们得到的是，当非线性术语$ f $仅满足某些增长条件，只有中心$ 0 $ and radius $ 4 $在一个圆圈中满足某些增长条件时，系统具有非平凡的解决方案$（u_λ，v_λ）$，带有$ \ |（u_λ，v_λ）\ | _____ {\ | _ {\ infty} \ le 2 $ for youftions $ a $ uptions $ upect $ utiest $ utiest $ utections $ utections utections $ ^ uptions $。满足$ \ |（u_λ，v_λ）\ | \ to 0 $ as $λ\ to \ infty $。此外，还获得了准椭圆方程的相应结果，这比椭圆系统的结果更好。
In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system \begin{eqnarray*} \begin{cases}{ccc} -\mbox{div}(ϕ_1(|\nabla u|)\nabla u)+V_1(x)ϕ_1(|u|)u=λF_u(x, u,v), \ \ x\in \mathbb R^N, -\mbox{div}(ϕ_2(|\nabla v|)\nabla v)+V_2(x)ϕ_2(|v|)v=λF_v(x, u,v), \ \ x\in \mathbb R^N, u\in W^{1,Φ_1}(\mathbb R^N), v\in W^{1,Φ_2}(\mathbb R^N), \end{cases} \end{eqnarray*} where $N\ge 2$, $\inf_{\mathbb R^N}V_i(x)>0,i=1,2$, and $λ>0$. We obtain that when the nonlinear term $F$ satisfies some growth conditions only in a circle with center $0$ and radius $4$, system has a nontrivial solution $(u_λ,v_λ)$ with $\|(u_λ,v_λ)\|_{\infty}\le 2$ for every $λ$ large enough, and the families of solutions $\{(u_λ,v_λ)\}$ satisfy that $\|(u_λ,v_λ)\|\to 0$ as $λ\to \infty$. Moreover, a corresponding result for a quasilinear elliptic equation is also obtained, which is better than the result for the elliptic system.
