论文标题
非线性kirchhoff方程的多峰值积极解决方案的存在
The existence of multi-peak positive solutions for nonlinear Kirchhoff equations
论文作者
论文摘要
在这项工作中，我们研究以下kirchhoff方程$ \ begin {case} - \ left（\ varepsilon^2 a +\ varepsilon b \ int _ {\ mathbb r^3} | \ nabla | \ nabla u |^2 \ right） {\ Mathbb {r}^{3}}，\\ u \ to 0，\ quad \ text {as} \ | x | \ to +\ to +\ to +\ ftty，\ end {cases} $ a，$ a，b> 0 $是常数，$ 2 <q <q <q <q <q <q <q <6 $，$ \ varepsilon> 0 $ as a parameter。根据对函数$ q（x）$的一些合适的假设，我们获得上面的方程式具有正面的多峰值解决方案，该解决方案集中在$ q（x）$的临界点，用于$ \ varepsilon> 0 $ 0 $，通过使用lyapunov-schmidt减少方法。我们将结果扩展到（离散连续的动力学系统6（2000），39--50）到非线性Kirchhoff方程。
In this work, we study the following Kirchhoff equation $$\begin{cases}-\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)Δu +u =Q(x)u^{q-1},\quad u>0,\quad x\in {\mathbb{R}^{3}},\\u\to 0,\quad \text{as}\ |x|\to +\infty,\end{cases}$$ where $a,b>0$ are constants, $2<q<6$, and $\varepsilon>0$ is a parameter. Under some suitable assumptions on the function $Q(x)$, we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of $Q(x)$ for $\varepsilon>0$ sufficiently small, by using the Lyapunov-Schmidt reduction method. We extend the result in (Discrete Contin. Dynam. Systems 6(2000), 39--50) to the nonlinear Kirchhoff equation.
