论文标题
具有渐近性3线性非线性的seminearschrödinger方程的鼻溶液
Nodal solutions for quasilinear Schrödinger equations with asymptotically 3-linear nonlinearity
论文作者
论文摘要
在本文中，我们关注的是susilinearearinearearirinearschrödinger方程\ begin {equation*}-Δu+v（x）u -u-uδ（u^2）= g（u），\ \ \ x \ in \ in \ mathbb {r}在无穷大处渐近3线性。对于$ \ inf _ {\ mathbb {r}^n} v> 0 $，我们显示了具有恰好一个节点的最少能量签名解决方案，对于任何整数$ k> 0 $，都有一对带有$ k $ nodes的签名更改的解决方案。此外，在$ \ inf _ {\ mathbb {r}^n} v = 0 $的情况下，上面的问题承认具有一个恰好一个节点的最少能量签名解决方案。该证明基于变异方法。特别是，引入了一些新的技巧和签名改变的Nehari歧管的方法，以克服因渐近3线性非线性出现而引起的困难。
In this paper, we are concerned with the quasilinear Schrödinger equation \begin{equation*} -Δu+V(x)u-uΔ(u^2)=g(u),\ \ x\in \mathbb{R}^{N}, \end{equation*} where $N\geq3$, $V$ is radially symmetric and nonnegative, and $g$ is asymptotically 3-linear at infinity. In the case of $\inf_{\mathbb{R}^N}V>0$, we show the existence of a least energy sign-changing solution with exactly one node, and for any integer $k>0$, there are a pair of sign-changing solutions with $k$ nodes. Moreover, in the case of $\inf_{\mathbb{R}^N}V=0$, the problem above admits a least energy sign-changing solution with exactly one node. The proof is based on variational methods. In particular, some new tricks and the method of sign-changing Nehari manifold depending on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically 3-linear nonlinearities.
