Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation
Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation
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DOI：
10.1007/s11784-016-0362-4
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发表时间：
2016-06
影响因子：
1.8
通讯作者：
G. Evéquoz;T. Weth
中科院分区：
文献类型：
--
作者：
G. Evéquoz;T. Weth
考虑非线性定常薛定谔方程$$\开始{aligned} -\Delta u -\lambda u= Q（x）|u| ^{p-2}u，\qquad \text {in }\mathbb {R}^N \end{aligned}$$在这种情况下，p是一个超线性的，次临界的指数，Q是一个有界的，非负的和非平凡的权函数，在和中有紧支持，是一个参数。在进一步的限制下，无论是对指数p还是对Q的形状，我们都建立了这个方程的非平凡解的连续分支的存在性，该分支对每个和都相交。这里，是一个显式的正常数，它只依赖于Nand。特别地，沿着分支的值的集合进入算子的本质谱。
We consider the nonlinear stationary Schrödinger equation $$\begin{aligned} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text {in }\mathbb {R}^N \end{aligned}$$in the case where,pis a superlinear, subcritical exponent,Qis a bounded, nonnegative and nontrivial weight function with compact support inandis a parameter. Under further restrictions either on the exponentpor on the shape ofQ, we establish the existence of a continuous branchof nontrivial solutions to this equation which intersectsfor everyand. Here,is an explicit positive constant which only depends onNand. In particular, the set of valuesalong the branch enters the essential spectrum of the operator.