一类非线性椭圆型方程单解的无穷多个整体连续分支
Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave-convex nonlinearity
摘要：
本文研究了环$Ω\subset\mathbb{R}^N$上半线性椭圆边值问题解的分支，其形式为\begin{align*} \begin{aligned} -Δu &= f_λ(|x|,u,|\nabla u|) &&\text{in }Ω, u &= 0 &&\text{on }\partialΩ, \end{aligned} \end{align*}，具有凹凸非线性项，一种特殊情况是Ambrosetti，Brezis和Cerami首先考虑的非线性项：$f_λ(|x|,u,|\nabla u|)=λ|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$ .虽然平凡解$u_0\equiv0$是非退化的，但若$λ=0$，我们证明了$(λ_0,u_0)=(0,0)$是分歧点.事实上，分歧的情况是非常奇异的：我们表明，有无穷多个全球连续的径向解决方案$\mathcal{C}_j^\pm\subset\mathbb{R}\times\mathcal{C}^1(Ω)$，$j\in\mathbb{N}_0$分叉从平凡的分支$\mathbb{R}\times\{0\}$在$(λ_0,u_0)=(0,0)$，并包括解决方案，具有精确的$j$节环。对这些连续统的详细研究表明，它们在$\mathbb{R}_{\ge0}\times\{0\}$处积累，因此每个带有$λ\ge0$的$(λ,0)$都是一个分叉点。此外，在$\mathcal{C}^1(Ω)$上增加无穷远处的一个点，它们也在$\mathbb{R}\times\{\infty\}$处累积，因此在每个$λ\in\mathbb{R}$处存在从无穷远处的分叉。
英文摘要：
We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -Δu &= f_λ(|x|,u,|\nabla u|) &&\text{in }Ω, u &= 0 &&\text{on }\partialΩ, \end{aligned} \end{align*} on an annulus $Ω\subset\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_λ(|x|,u,|\nabla u|)=λ|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial solution $u_0\equiv0$ is nondegenerate if $λ=0$ we prove that $(λ_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifurcation scenario is very singular: We show that there are infinitely many global continua of radial solutions $\mathcal{C}_j^\pm\subset\mathbb{R}\times\mathcal{C}^1(Ω)$, $j\in\mathbb{N}_0$ which bifurcate from the trivial branch $\mathbb{R}\times\{0\}$ at $(λ_0,u_0)=(0,0)$ and consist of solutions having precisely $j$ nodal annuli. A detailed study of these continua shows that they accumulate at $\mathbb{R}_{\ge0}\times\{0\}$ so that every $(λ,0)$ with $λ\ge0$ is a bifurcation point. Moreover, adding a point at infinity to $\mathcal{C}^1(Ω)$ they also accumulate at $\mathbb{R}\times\{\infty\}$, so there is bifurcation from infinity at every $λ\in\mathbb{R}$.
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