Global bifurcation for quasilinear elliptic equations on $mathbb{R}^{N}$
Global bifurcation for quasilinear elliptic equations on
$mathbb{R}^{N}$
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$mathbb{R}^{N}$ 上拟线性椭圆方程的全局分岔
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
C. Stuart
中科院分区：
文献类型：
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作者：
P. Rabier;C. Stuart
抽象的。在本文中，我们讨论了一些连通解集的全局行为
一类广义二阶拟线性椭圆型方程的$(lambda，u)$
Egin{公式}-sum_{α，eta=1}^{N}a_{alphaeta}(x，u(X)，
Abla u(X))Partial_{Alpha}Partial_{Eta}u(X)+b(x，u(X)，
Abla u(X)，lambda)=0结束{公式}
为
$xinmathbb{R}^{N}$其中
$lambda$是实参数，函数u需要满足条件
Egin{公式}LimLimits_{Left|x
夜晚|
Ightarrowinfty}u(X)=0。结束{公式}
基本工具是由Fitzpatrick，Pejsachowicz和Rabier得到的指数为零的适当Fredholm型映射的次数。要使用这个程度，问题必须以以下形式表示
$F：J IMES X
右行Y$其中J是区间，X和Y是Banach空间，F是
$C^{1}$映射是Fredholm的，在闭有界子集上是真的。我们用的是通常的空间
$X=W^{2，p}(mathbb{R}^{N})$AND
$Y=L^{p}(mathbb{R}^{N})$。那么主要的困难就是找出
确保F的适定性的$a_{alphaeta}$和b。我们对此的方法是基于最近的一些工作，其中，在假设
$a_{alphaeta}$和b在x中渐近周期为$Left|x
夜晚|
，我们得到了简单的条件，这些条件是充分必要条件
$F(lambda，CDOT)：X
证明了在X的有界闭子集上Y$是Fredholm型的，并且是真的，特别是渐近问题在X中非零解的不存在性在这个问题中起着至关重要的作用。我们的结果建立了一般问题解的全局分支的分支。文中还讨论了各种特殊情况。即使对于以下形式的半线性方程
[-Delta u(X)+f(x，u(X))=lambda u(X)，]
我们的结果涵盖了文献中其他方法范围之外的情况。
Abstract. In this paper we discuss the global behaviour of some connected sets of solutions
$(lambda,u)$ of a broad class of second order quasilinear elliptic equations
egin{equation} -sum_{alpha,eta=1}^{N}a_{alphaeta}(x,u(x),
abla u(x))partial_{alpha }partial_{eta}u(x)+b(x,u(x),
abla u(x),lambda)=0 end{equation}
for
$xinmathbb{R}^{N}$ where
$lambda$ is a real parameter and the function u is required to satisfy the condition
egin{equation} limlimits_{left| x
ight|
ightarrowinfty}u(x)=0. end{equation}
The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form
$F:J imes X
ightarrow Y$ where J is an interval, X and Y are Banach spaces and F is a
$C^{1}$ map which is Fredholm and proper on closed bounded subsets. We use the usual spaces
$X=W^{2,p}(mathbb{R}^{N})$ and
$Y=L^{p}(mathbb{R}^{N})$. Then the main difficulty involves finding general conditions on
$a_{alphaeta}$ and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that
$a_{alphaeta} $ and b are asymptotically periodic in x as $left| x
ight|
ightarrowinfty$, we have obtained simple conditions which are necessary and sufficient for
$F(lambda,cdot):X
ightarrow Y$ to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form
[ -Delta u(x)+f(x,u(x))=lambda u(x), ]
our results cover situations outside the scope of other methods in the literature.