arXiv 1106.58792011-06-29math.AP

渐近线性薛定谔方程的全局分支

Global bifurcation for asymptotically linear Schrödinger equations

François Genoud

发表信息：: NoDEA Nonlinear Differential Equations Appl. 20 (2013), 23-35
DOI：: 10.1007/s00030-012-0152-7

摘要：

我们证明了一类非常一般的渐近线性薛定谔方程的全局渐近分支\begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation}的方法是拓扑的，基于度理论的最新发展。我们在适当的索伯列夫空间$X=W^{2,p}({\mathbb R}^N)$中利用反演$u\to v:= u/\Vert u\Vert_X^2$，首先得到了一个辅助问题在变量$(λ,v) \in {\mathbb R} \x X$中平凡解的分支.这个问题缺乏紧凑性和规律性，需要截断过程。回到原来的问题，我们得到了正/负解的全局分支“从无穷大分支”。我们认为，对于我们的分歧方法所覆盖的$λ$的值，我们得到的\eqref{1}正解的存在性结果是迄今为止最一般的

英文摘要：

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schrödinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} The method is topological, based on recent developments of degree theory. We use the inversion $u\to v:= u/\Vert u\Vert_X^2$ in an appropriate Sobolev space $X=W^{2,p}({\mathbb R}^N)$, and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables $(λ,v) \in {\mathbb R} \x X$. This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions 'bifurcating from infinity'. We believe that, for the values of $λ$ covered by our bifurcation approach, the existence result we obtain for positive solutions of \eqref{1} is the most general so far

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