论文标题
具有关键生长非线性的Kirchhoff类型方程的归一化解决方案
Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity
论文作者
论文摘要
在本文中，我们关注Kirchhoff类型方程的归一化解决方案 \ begin {equination*} -m \ left（\ int _ {\ r^n} | \ nabla u |^2 \ mathrm {d} x \ right）Δu=λu=λu +f（u +f（u） S_C：= \ left \ {当$ n = 2 $和$ f $在无穷大时具有指数级的临界增长时，可以通过变异方法获得归一化山通道类型解决方案。当$ n \ ge 4 $，$ m（t）= a+bt $带有$ a $，$ b> 0 $和$ f $的INFINITY在INFINITY时具有临界增长，我们调查了归一化基态解决方案和标准化的山通行证型解决方案的存在。此外，还考虑了归一化解决方案的不存在。
In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)Δu = λu +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered.
