论文标题
关于$（p，q）$ - 拉普拉斯系统的三种非负解决方案
On the existence of three non-negative solutions for a $(p,q)$-Laplacian system
论文作者
论文摘要
本文研究了\ begin {equation*}（\ Mathcal {p}）\ left \ {\ oken {aligned}（-Δ）^{s_1} _ {p_1} _ {p_1} u＆=＆= \ la f_1 \ f_1 \，（x，U，V） \ om，\\（-Δ）^{s_2} _ {p_2} v＆= \ la f_2 \，（x，x，u，u，v） +g_2（x，x，v）\，\，\ mbox {in} \，\ om，\ om，\ om，\\ om，\ \ = v＆= v＆= v＆= 0 \，\，\，\，\，\，\，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c \ end {Aligned} \ right。 \ end {equation*}其中$ \ om \ subset \ rn $是一个平稳的边界域，$ s_1，$ s_1，s_2 \ in（0,1）$，$ 1 <p_i <\ frac {n} {n} {s_i} $，$ i = 1,2 $ i = 1,2 $ f_i $和$ i = $ g_i $ i = 1.我们证明，使用变异方法，在$λ$的限制范围内，至少存在$（\ Mathcal P）$的至少三个非负解决方案。结果，我们还得出结论，当我们考虑更通用的非本地操作员$ \ MATHCAL L_ {ϕ_I} $而不是$（ - δ）^{s_i} _ {p_i} $ in $（\ Mathcal p）$中时，可以获得类似的结果。
The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-Δ)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-Δ)^{s_2}_{p_2} v &=\la f_2\,(x,u,v) +g_2(x,v) \,\mbox{ in }\, \Om, \\ u=v &= 0 \,\mbox{in }\, \Rn \setminus \Om, \\ \end{aligned} \right. \end{equation*} where $\Om \subset \Rn$ is a smooth bounded domain with smooth boundary, $s_1,s_2 \in (0,1)$, $1<p_i<\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\mathcal P)$ under restrictive range of $λ$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator $\mathcal L_{ϕ_i}$ instead of $(-Δ)^{s_i}_{p_i}$ in $(\mathcal P)$.
