论文标题
$ \ mathbb r^n \ setMinus \ {0 \} $具有耐力的非线性椭圆方程的鲜明存在和分类结果
Sharp existence and classification results for nonlinear elliptic equations in $\mathbb R^N\setminus\{0\}$ with Hardy potential
论文作者
论文摘要
对于$ n \ geq 3 $，由布雷兹斯和维伦的开创性论文（Arch。Mech。Anal。75（1）：1--6，1980/81），没有$-ΔU+u^Q = 0 $ in $ \ \ \ \ \ \ \ \ Mathbb r^n n \ setMinus \ setminus \ \ \ geq n If $ q的$-ΔU+u^q = 0 $ in的正面解决方案；对于$ 1 <q <n/（n-2）$，所有正面$ c^1的存在和配置文件（\ mathbb r^n \ setMinus \ {0 \}）$ solutions由弗里德曼（Friedman）和Véron（Arch。Arch。ComationMech。Anal。96（4）（4）（4）：359---387，1986）。 在本文中，对于每一个$ q>> 1 $和$θ\ in \ mathbb r $，我们证明了非线性椭圆问题（*）$-ΔU-Δu-λ\，| x |^{ - 2} \，U+| x | x |^θu^q = 0 $ $ c^1（\ mathbb r^n \ setMinus \ {0 \}）$解决方案时，仅当$λ>λ^*$，其中$λ^*=θ（n-2-θ）$带有$θ=（θ+2）/（q-1）/（q-1）$。我们表明（a）如果$λ>（n-2）^2/4 $，则$ u_0（x）=（λ-λ^*） $ u_0 \ cup \ {u_ {γ，q，λ}：\γ\ in（0，\ infty）\} $。我们给出了$ u_ {γ，q，λ} $接近零和无限的精确行为，区分$ 1 <q <q <q <q_ {n，θ} $和$ q>> \ max \ max \ {q_ {q_ {n，θ}，1 \}，1 \}，1 \} $，其中$ q_ {n，θ}} n，θ}} =（N+2θ+2+2）$（n+2θ+2）$（n+2θ+2）$。 此外，对于$θ\ leq -2 $，我们解决了$ω\ setMinus \ {0 \ {0 \} $的所有积极解决方案的结构，但要符合$ u | _ {\partialΩ} = 0 $，$ u | _ {\partialΩ} = 0 $，其中$ω$是一个平滑的界面，是一个平滑的界面，包含了cornece n e cornece of cornece of cornece of cornece of cornece of Soce。 2014年）和Wei-du（J.微分方程262（7）：3864--3886，2017）。
For $N\geq 3$, by the seminal paper of Brezis and Véron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-Δu+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $1<q<N/(N-2)$ the existence and profiles near zero of all positive $C^1(\mathbb R^N\setminus \{0\})$ solutions are given by Friedman and Véron (Arch. Rational Mech. Anal. 96(4):359--387, 1986). In this paper, for every $q>1$ and $θ\in \mathbb R$, we prove that the nonlinear elliptic problem (*) $-Δu-λ\,|x|^{-2}\,u+|x|^θu^q=0$ in $\mathbb R^N\setminus \{0\}$ with $u>0$ has a $C^1(\mathbb R^N\setminus \{0\})$ solution if and only if $λ>λ^*$, where $λ^*=Θ(N-2-Θ) $ with $Θ=(θ+2)/(q-1)$. We show that (a) if $λ>(N-2)^2/4$, then $U_0(x)=(λ-λ^*)^{1/(q-1)}|x|^{-Θ}$ is the only solution of (*) and (b) if $λ^*<λ\leq (N-2)^2/4$, then all solutions of (*) are radially symmetric and their total set is $U_0\cup \{U_{γ,q,λ}:\ γ\in (0,\infty) \}$. We give the precise behavior of $ U_{γ,q,λ}$ near zero and at infinity, distinguishing between $1<q<q_{N,θ}$ and $q>\max\{q_{N,θ},1\}$, where $q_{N,θ}=(N+2θ+2)/(N-2)$. In addition, for $θ\leq -2$ we settle the structure of the set of all positive solutions of (*) in $Ω\setminus \{0\}$, subject to $u|_{\partialΩ}=0$, where $Ω$ is a smooth bounded domain containing zero, complementing the works of C\^ırstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).
