论文标题
kirchhoff类型$（φ_1，φ_2）$ - laplacian系统具有本地超级线性条件的Laplacian System的存在和多样性
Existence and multiplicity of solutions for a class of Kirchhoff type $(Φ_1,Φ_2)$-Laplacian system with locally super-linear condition in $\mathbb{R}^N$
论文作者
论文摘要
我们研究了整个空间上的非线性kirchhoff类型quasilinear椭圆系统的弱解决方案的存在和多样性。我们假设非线性术语满足本地超级 - $（m_1，m_2）$状态，即$ \ lim_ {|（u，v）| \ rightArrow+\ rightarrow+\ infty} \ frac {f（x，x，x，x，u，u，v）}} } x \在g $中，其中$ g $是$ \ mathbb {r}^n $中的一个域，它比著名的ambrosseti-rabinowitz条件弱，并且自然全球限制， $ \ lim_ {|（u，v）| \ rightarrow+\ iftty} \ frac {f（x，x，u，u，v）} {| u |^{m_1}+| v | v |^{m_2}} =++\ iffty \ mbox {for A.E. } x \ in \ mathbb {r}^n $。我们通过使用经典的山间通过定理获得该系统至少具有一个弱解决方案。在一定程度上，我们的定理扩展了Tang-Yu的结果[动力学和微分方程杂志，2019，31（1）：369-383]。此外，在上述自然限制下，我们通过使用对称的山地通定理具有无限的高能解决方案，这与Wang-Zhang-fang的结果不同[非线性科学与应用杂志，2017，1017，10（7）：3792-3814]，即使我们考虑在界面上的系统条件，也要在界面上进行界面。
We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$ condition, that is, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in \mathbb{R}^N$. We obtain that system has at least one weak solution by using the classical Mountain Pass Theorem. To a certain extent, our theorems extend the results of Tang-Lin-Yu [Journal of Dynamics and Differential Equations, 2019, 31(1): 369-383]. Moreover, under the above naturally global restriction, we obtain that system has infinitely many weak solutions of high energy by using the Symmetric Mountain Pass Theorem, which is different from those results of Wang-Zhang-Fang [Journal of Nonlinear Sciences and Applications, 2017, 10(7): 3792-3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.
