本文研究如下带有奇异位势的矩阵型强奇异偏微分方程

$ \left\{ {\begin{array}{*{20}{l}} { - {\mathop{\rm div}\nolimits} (M(x)\nabla u) = |x{|^{ - \mu }}{u^{ - p}}}&{{\rm{ in }}\ \Omega ,}\\ {u > 0}&{{\rm{ in }}\ \Omega ,}\\ {u = 0}&{{\rm{ on }}\ \ \partial \Omega ,} \end{array}} \right. $

(1)

其中，0∈Ω是$\mathbb{R}$ⁿ(n≥3)中具有光滑边界的有界开集，M(x)是定义在Ω上的实对称矩阵, 满足：存在正常数α, β使得M(x)ξ·ξ≥α|ξ|², |detM(x)|≤β, ∀ξ∈$\mathbb{R}$ⁿ, ∀x∈Ω，-3 < -p < -1, -n < -μ < 0。

上述方程的一般形式是-div(M(x)$\nabla $u)=f(x, u)。当x→0，f(x, ·)→∞时，称此类偏微分方程是带有奇异位势的偏微分方程。当s→0, f(·, s)→∞时，称此类偏微分方程是具有奇性的偏微分方程。特别地，若有s→0, f(·, s)s→∞时，称此类偏微分方程是具有强奇性的偏微分方程。具有奇性的偏微分方程广泛应用于多种物理模型中，比如流体力学中的边界层现象、化学异构催化剂、冰川移动等，详情可参考文献[1-2]。近年来，学者们对这一类型方程做出过一系列研究工作, 并得到了相当丰富且深刻的研究成果。Lazer和McKenna^[3]对基本形式方程-Δu=p(x)u^-p进行研究，并得到结论：若p(x)∈C^α(Ω)(0 < α<1), p(x)>0, ∀x∈Ω, 则对∀-p < 0, 方程存在唯一解u_-p∈C^2+α(Ω)∩C¹(Ω), 且若-p < -1, 则u_-p∉C¹(Ω), 以及u_-p∈H₀¹(Ω)当且仅当-p>-3。Boccardo和Orsina^[4]首先研究矩阵型奇异偏微分方程，他们指出当-p < -1，非负系数函数p(x)∈L¹(Ω)时，方程存在解u∈H_loc¹(Ω), ${u^{\frac{{p + 1}}{2}}} \in H_0^1\left(\Omega \right)$, 但不能得到u∈H₀¹(Ω)。而对于带有奇异位势椭圆型偏微分方程，Bae和Pahk^[5]运用Brezis-Nirenberg^[6]的方法，研究一类带有奇异位势的奇性椭圆型偏微分方程，它的正径向解和节点径向解的存在性取决于方程中指数所属范围。Caldiroli和Musina^[7-8]研究在Sobolev嵌入中紧性可能缺失的情况下，一类带有奇异位势的二维奇性椭圆型偏微分方程的正解的存在性对所给区域的要求。Ruiz和Willem^[9]运用Palais^[10]的方法，研究一类带有Sobolev临界指数和奇异位势的非线性椭圆问题的正解的存在性。

1 本文主要研究成果

众所周知(参见文献[7])，如下问题：

$ \left\{ {\begin{array}{*{20}{l}} { - \Delta u = |x{|^{ - \mu }}{u^q}}&{{\rm{ in }}\ \Omega ,}\\ {u > 0}&{{\rm{ in }}\ \Omega ,}\\ {u = 0}&{{\rm{ on }}\ \partial \Omega ,} \end{array}} \right. $

不存在正解，这里0∈Ω是$\mathbb{R}$ⁿ(n≥2)中的开集，q>1, -μ≤-2，也就是说-2是正解存在的临界值。而在我们的前期工作^[11]中指出，对于带有奇异位势的负指数方程而言，-2不再是正解存在的临界值。且在文献[12]中证得，当-3 < -p < -1, -n≤-μ≤0时，方程(1)存在H₀¹(Ω)正解。本文将进一步研究解的性质：无界性及逼近速度。以下是本文的主要研究成果：

定理1.1  设Ω是$\mathbb{R}$ⁿ(n≥3)中包含原点的具有光滑边界的有界开集, M(x)是定义在Ω上的实对称矩阵，满足：存在正常数α, β使得M(x)ξ·ξ≥α|ξ|², |detM(x)|≤β, ∀ξ∈$\mathbb{R}$ⁿ, ∀x∈Ω, -3 < -p < -1, -n < -μ < 0, 则方程

$ \left\{ {\begin{array}{*{20}{l}} { - {\mathop{\rm div}\nolimits} (\mathit{\boldsymbol{M}}(x)\nabla u) = |x{|^{ - \mu }}{u^{ - p}}}&{{\rm{ in }}\ \Omega ,}\\ {u > 0}&{{\rm{ in }}\ \Omega ,}\\ {u = 0}&{{\rm{ on }}\ \partial \Omega ,} \end{array}} \right. $

存在H₀¹(Ω)-解u_-p。

定理1.2  设Ω是$\mathbb{R}$ⁿ(n≥3)中包含原点的具有光滑边界的有界开集, M(x)是定义在Ω上的实对称矩阵，满足：存在正常数α, β使得M(x)ξ·ξ≥α|ξ|², |detM(x)|≤β, ∀ξ∈$\mathbb{R}$ⁿ, ∀x∈Ω, -3 < -p < -1, -n < -μ < -1-$\frac{n}{2}$, 则方程(1)的任一H₀¹-解u无界，即u∉L^∞(Ω)。

定理1.3  设Ω是$\mathbb{R}$ⁿ(n≥3)中包含原点的具有光滑边界的有界开集, -3 < -p < -1, 如果-n < -μ < -2, 则方程

$ \left\{ {\begin{array}{*{20}{l}} { - \Delta u = |x{|^{ - \mu }}{u^{ - p}}}&{{\rm{ in }}\ \Omega ,}\\ {u > 0}&{{\rm{ in }}\ \Omega ,}\\ {u = 0}&{{\rm{ on }}\ \partial \Omega ,} \end{array}} \right. $

(2)

对∀$0 < l < \frac{{\mu - 2}}{{p + 1}}$, C²(Ω\{0})-解u都满足性质：

$ \mathop {{\rm{limsup}}}\limits_{x \to 0} \frac{{u(x)}}{{|x{|^{ - l}}}} = + \infty . $

注:定理1.1的证明可参考文献[12]。在定理1.1研究的一般问题中，参数μ满足-n < -μ < -1-$\frac{n}{2}$时，得到正解的无界性。而当M(x)≡I，参数μ满足-n < -μ < -2时，正解就会是无界解, 具体可参考文献[11](Theorem 1.3)。

定义H₀¹(Ω)中范数为

$ \left\| u \right\| = {\left( {\int_\Omega | \nabla u{|^2}{\rm{d}}x} \right)^{\frac{1}{2}}},\forall u \in H_0^1(\Omega ), $

称u是方程(1)的H₀¹(Ω)-解, 如果u∈H₀¹(Ω), u>0 a.e. in Ω, 满足∀φ∈H₀¹(Ω),

$ \int_\Omega \mathit{\boldsymbol{M}} (x)\nabla u \cdot \nabla \varphi {\rm{d}}x - \int_\Omega | x{|^{ - \mu }}{u^{ - p}} \cdot \varphi {\rm{d}}x = 0. $

2 定理1.2的证明

用反证法证明。假设u∈L^∞(Ω), 即$\mathop {{\text{ess sup}}u}\limits_\Omega $ < +∞, 且设δ为充分小的正数，B_2δ(0)是以原点为球心，以2δ为半径包含在Ω里的闭球。构造截断函数ϕ_δ，满足条件：ϕ_δ∈C₀^∞(Ω), 在Ω₀=B_2δ(0)\B_δ(0)内，0≤ϕ_δ≤1, 在${\Omega _1} = {B_{\frac{5}{3}\delta }}\left(0 \right)\backslash {B_{\frac{4}{3}\delta }}\left(0 \right)$内，ϕ_δ≡1, 在Ω\Ω₀内，ϕ_δ≡0, 且

$ \left| {{D^\beta }{\phi _\delta }} \right| \le \frac{{c(n,|\beta |)}}{{{\delta ^{|\beta |}}}}, $

其中β=(β₁, β₂, …, β_n)为多元指标，|β|=β₁+…+β_n, c(n, |β|)是与n, β有关的正参数，特别地，

$ \left| {\nabla {\phi _\delta }} \right| \le \frac{{c(n)}}{\delta }. $

由于u是方程(1)的解，有

$ \int_\Omega \mathit{\boldsymbol{M}} (x)\nabla u \cdot \nabla {\varphi _\delta }{\rm{d}}x = \int_\Omega | x{|^{ - \mu }}{u^{ - p}}{\phi _\delta }{\rm{d}}x. $

(3)

由|M(x)ξ·γ|≤$\frac{\beta }{{{\alpha ^{n - 1}}}}$|ξ||γ|, ∀x∈Ω, ∀ξ, γ∈$\mathbb{R}$ⁿ(证明可参考文献[12]中引理1)，估计(3)左边值，有

$ \begin{array}{l} \int_\Omega {\boldsymbol{M}} (x)\nabla u \cdot \nabla {\phi _\delta }{\rm{d}}x \le \frac{\beta }{{{\alpha ^{n - 1}}}}\int_\Omega | \nabla u| \cdot \left| {\nabla {\phi _\delta }} \right|{\rm{d}}x\\ \ \ \ \ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le \frac{\beta }{{{\alpha ^{n - 1}}}} \cdot \frac{{c(n)}}{\delta }\int_{{\Omega _0}} | \nabla u|{\rm{d}}x\\ \ \ \ \ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le \frac{\beta }{{{\alpha ^{n - 1}}}} \cdot \frac{{c(n)}}{\delta }{\left( {\int_{{\Omega _0}} {{\rm{d}}} x} \right)^{\frac{1}{2}}} \cdot \\ \ \ \ \ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left( {\int_{{\Omega _0}} | \nabla u{|^2}{\rm{d}}x} \right)^{\frac{1}{2}}}\\ \ \ \ \ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {c_1}{\delta ^{\frac{n}{2} - 1}}{\left( {\int_{{\Omega _0}} | \nabla u{|^2}{\rm{d}}x} \right)^{\frac{1}{2}}}, \end{array} $

其中c₁是与α, β, n有关、与δ无关的正数值。估计(3)右边值，有

$ \begin{array}{l} \int_\Omega | x{|^{ - \mu }}{u^{ - p}}{\phi _\delta }{\rm{d}}x \ge \int_{{\Omega _1}} | x{|^{ - \mu }}{u^{ - p}}{\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ge {\left( {\mathop {{\mathop{\rm esssup}\nolimits} u}\limits_\Omega } \right)^{ - p}}\int_{{\Omega _1}} | x{|^{ - \mu }}{\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\left( {\mathop {{\mathop{\rm esssup}\nolimits} u}\limits_\Omega } \right)^{ - p}}{c_2}{\delta ^{n - \mu }}, \end{array} $

其中c₂是与α, β, n有关、与δ无关的正数值。于是得到

$ {c_1}{\delta ^{\frac{n}{2} - 1}}{\left( {\int_{{\Omega _0}} | \nabla u{|^2}{\rm{d}}x} \right)^{\frac{1}{2}}}{(\mathop {{\mathop{\rm esssup}\nolimits} u}\limits_\Omega )^{ - p}}{c_2}{\delta ^{n - \mu }}, $

即

$ \begin{array}{l} {c_1}\left\| u \right\|{(\mathop {{\mathop{\rm esssup}\nolimits} }\limits_\Omega u)^p} \ge {c_2}{\delta ^{n - \mu - \frac{n}{2} + 1}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {c_2}{\delta ^{\frac{n}{2} + 1 - \mu }},\quad \forall 0 < \mu < n. \end{array} $

当-μ < -$\frac{n}{2}$-1时，随着δ→0, 上式右边→+∞, 左边为有限数，产生矛盾，因此有结论：当-n < -μ < -$\frac{n}{2}$-1时，方程(1)的解无界，而且是当|x|→0时，解u是无界的。

3 定理1.3的证明

用反证法证明。假设方程(2)的C²(Ω\{0})-解u不满足性质

$ \mathop {{\rm{limsup}}}\limits_{x \to 0} \frac{{u(x)}}{{|x{|^{ - l}}}} = + \infty ,\forall 0 < l < \frac{{\mu - 2}}{{p + 1}}. $

则存在$l \in \left({0, \frac{{\mu - 2}}{{p + 1}}} \right)$, 对以r(充分小)为半径、以0为圆心的闭球B_r(0)⊂Ω, 有u(x)≤c|x|^-l, ∀x∈B_r(0), c为某一正常数。由解u∈C²(Ω\{0}), 知道

$ - \Delta u \equiv |x{|^{ - \mu }}{u^{ - p}},\forall x \in \Omega \backslash \{ 0\} . $

(4)

令R=r^2-n, 定义区域

$ {\Omega (t) = \left\{ {x \in {B_r}(0):{t^{\frac{1}{{2 - n}}}} \le |x| \le r} \right\},} $

$ {\Gamma (t) = \left\{ {x \in {B_r}(0):|x| = {t^{\frac{1}{{2 - n}}}}} \right\},} $

其中t∈(R, +∞).取Ω(t)的外侧为正向, n表示∂Ω(t)的单位外法向量，即在边界Γ(R)中，n是指向背离原点方向的单位法向量，在边界Γ(t)中，n是指向原点方向的单位法向量。令Γ(t)上的点x的参数表达式为$x = ({x_1}, \ldots, {x_n}) = $$({t^{\frac{1}{{2 - n}}}}{\text{cos}}{\theta _1}, {t^{\frac{1}{{2 - n}}}}{\text{sin}}{\theta _1}{\text{cos}}{\theta _2}, \ldots, {t^{\frac{1}{{2 - n}}}}{\text{sin}}{\theta _1} \ldots {\text{cos}}{\theta _{n - 1}}, {t^{\frac{1}{{2 - n}}}}{\text{sin}}{\theta _1} \ldots {\text{sin}}{\theta _{n - 1}})$, 其中，t, θ₁, …, θ_n-1为参数，0≤θ_i < π, i=1, …, n-2, 0≤θ_n-1 < 2π。则在Γ(t)中，有

$ {\mathit{\boldsymbol{n}} = \left( { - \cos {\theta _1}, \cdots , - \sin {\theta _1} \cdots \sin {\theta _{n - 1}}} \right) = - \frac{x}{{|x|}},} $

$ {x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right) \cdot \mathit{\boldsymbol{n}} = - {t^{\frac{1}{{2 - n}}}} = - |x|.} $

考虑函数g(x)=$\frac{{\text{1}}}{{n - 2}}$|x|^2-n, x∈$\mathbb{R}$ⁿ\{0}, 易知g(x)∈C^∞(Ω(t)), 且有

$ \begin{array}{l} \nabla g(x) = \left( { - {x_1}|x{|^{ - n}}, \cdots , - {x_n}|x{|^{ - n}}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = - x|x{|^{ - n}},\forall x \in \Omega (t), \end{array} $

$ |\nabla g(x)| = |x{|^{1 - n}},\forall x \in \Omega (t), $

$ \begin{array}{l} \frac{{\partial g}}{{\partial \mathit{\boldsymbol{n}}}} = \nabla g \cdot \mathit{\boldsymbol{n}} = |x{|^{ - n}}( - x) \cdot \mathit{\boldsymbol{n}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = |x{|^{1 - n}} = |\nabla g|,\forall x \in \Gamma (t), \end{array} $

$ \Delta g(x) = 0,\forall x \in \Omega (t). $

由格林恒等式，有

$ \int_{\Omega (t)} \nabla u \cdot \nabla g{\rm{d}}x + \int_{\Omega (t)} u \Delta g{\rm{d}}x = \int_{\partial \Omega (t)} u \frac{{\partial g}}{{\partial \mathit{\boldsymbol{n}}}}{\rm{d}}\sigma , $

即

$ \begin{array}{l} \int_{\Omega (t)} \nabla u \cdot \nabla g{\rm{d}}x = \int_{\partial \Omega (t)} u \frac{{\partial g}}{{\partial \mathit{\boldsymbol{n}}}}{\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \int_{\Gamma (R)} u \frac{{\partial g}}{{\partial \mathit{\boldsymbol{n}}}}{\rm{d}}\sigma + \int_{\Gamma (t)} u |\nabla g|{\rm{d}}\sigma . \end{array} $

其中dσ表示∂Ω(t)中的n-1维单位面积元。令A(t)=∫_Γ(t)u|$\nabla $g|dσ, 则

$ A(t) = \int_{\Omega (t)} \nabla u \cdot \nabla g{\rm{d}}x - \int_{\Gamma (R)} u \frac{{\partial g}}{{\partial \mathit{\boldsymbol{n}}}}{\rm{d}}\sigma . $

计算A(t)的一阶导数，

$ \begin{array}{l} {A^\prime }(t) = \frac{{{\rm{d}}A(t)}}{{{\rm{d}}t}} = \frac{{\rm{d}}}{{{\rm{d}}t}}\int_{\Omega (t)} \nabla u \cdot \nabla g{\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \int_{\partial \Omega (t)} {(\nabla u \cdot \nabla g)} \left( {\frac{{\partial x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right)}}{{\partial t}} \cdot \mathit{\boldsymbol{n}}} \right){\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \int_{\Gamma (t)} {(\nabla u \cdot \nabla g)} \left( {\frac{{\partial x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right)}}{{\partial t}} \cdot \mathit{\boldsymbol{n}}} \right){\rm{d}}\sigma , \end{array} $

在Γ(t)中，有

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\partial x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right)}}{{\partial t}}\\ = \left( {\frac{1}{{2 - n}}{t^{\frac{{n - 1}}{{2 - n}}}}\cos {\theta _1}, \cdots ,\frac{1}{{2 - n}}{t^{\frac{{n - 1}}{{2 - n}}}}\sin {\theta _1} \cdots \sin {\theta _{n - 1}}} \right), \end{array} $

$ \mathit{\boldsymbol{n}} \cdot \frac{{\partial x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right)}}{{\partial t}} = \frac{1}{{n - 2}}{t^{\frac{{n - 1}}{{2 - n}}}}. $

则

$ \begin{array}{l} {A^\prime }(t) = \frac{1}{{n - 2}}{t^{\frac{{n - 1}}{{2 - n}}}}\int_{\Gamma (t)} {(\nabla u \cdot \nabla g)} {\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { = \frac{1}{{n - 2}}{t^{\frac{{n - 1}}{{2 - n}}}}\int_{\Gamma (t)} - |x{|^{ - n}}x \cdot \nabla u{\rm{d}}\sigma }\\ { = \frac{1}{{n - 2}}\int_{\Gamma (t)} - \frac{x}{{|x|}} \cdot \nabla u{\rm{d}}\sigma }\\ { = \frac{1}{{n - 2}}\int_{\Gamma (t)} \mathit{\boldsymbol{n}} \cdot \nabla u{\rm{d}}\sigma }\\ { = \frac{1}{{n - 2}}\int_{\Gamma (t)} {\frac{{\partial u}}{{\partial \mathit{\boldsymbol{n}}}}} {\rm{d}}\sigma }\\ { = \frac{1}{{n - 2}}\left( {\int_{\Omega (t)} \Delta u{\rm{d}}x - \int_{\Gamma (R)} {\frac{{\partial u}}{{\partial \mathit{\boldsymbol{n}}}}} {\rm{d}}\sigma } \right)} \end{array} \end{array} $

上述等式最后一步根据散度定理而来。计算A(t)的二阶导数，有

$ \begin{array}{l} {A^{\prime \prime }}(t) = \frac{1}{{n - 2}}\int_{\partial \Omega (t)} \Delta u \cdot \left( {\frac{{\partial x\left( {t,{\theta _1}, \cdots ,{\theta _{n - 1}}} \right)}}{{\partial t}} \cdot \mathit{\boldsymbol{n}}} \right){\rm{d}}x\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { = \frac{1}{{n - 2}}\int_{\Gamma (t)} \Delta u \cdot \frac{1}{{n - 2}}{t^{\frac{{n - 1}}{{2 - n}}}}{\rm{d}}x}\\ { = {{\left( {\frac{1}{{n - 2}}} \right)}^2}{t^{\frac{{n - 1}}{{2 - n}}}}\int_{\Gamma (t)} \Delta u{\rm{d}}\sigma .} \end{array} \end{array} $

再根据对解u的假设，u(x)≤c|x|^-l, ∀x∈B_R(0), $l \in \left({0, \frac{{\mu - 2}}{{p + 1}}} \right)$和式(4), 以及关系式

$ \begin{array}{l} |\Gamma (t)| = \int_{\Gamma (t)} {\rm{d}} \sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { \le {{\left( {\int_{\Gamma (t)} {{u^{\frac{p}{{1 + p}}\frac{{1 + p}}{p}}}} {\rm{d}}\sigma } \right)}^{\frac{p}{{p + 1}}}}{{\left( {\int_{\Gamma (t)} {{u^{\frac{{ - p}}{{1 + p}} \cdot (1 + p)}}} {\rm{d}}\sigma } \right)}^{\frac{1}{{p + 1}}}}}\\ { = {{\left( {\int_{\Gamma (t)} u {\rm{d}}\sigma } \right)}^{\frac{p}{{1 + p}}}}{{\left( {\int_{\Gamma (t)} {{u^{ - p}}} {\rm{d}}\sigma } \right)}^{\frac{1}{{1 + p}}}},}\\ { \Rightarrow \int_{\Gamma (t)} {{u^{ - p}}} {\rm{d}}\sigma \ge |\Gamma (t){|^{1 + p}}{{\left( {\int_{\Gamma (t)} u {\rm{d}}\sigma } \right)}^{ - p}},} \end{array} \end{array} $

我们来估算A(t), A″(t)。

$ \begin{array}{l} A(t) = \int_{\Gamma (t)} u |\nabla g|{\rm{d}}\sigma = \int_{\Gamma (t)} u |x{|^{1 - n}}{\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le c\int_{\Gamma (t)} | x{|^{1 - n - l}}{\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = c \cdot {t^{\frac{{1 - n - l}}{{2 - n}}}}|\Gamma (t)| = {d_1}{t^{\frac{{ - l}}{{2 - n}}}}, \end{array} $

$ \begin{array}{l} {A^{\prime \prime }}(t) = {\left( {\frac{1}{{n - 2}}} \right)^2}{t^{\frac{{n - 1}}{{2 - n}}}}\int_{\Gamma (t)} \Delta u{\rm{d}}\sigma \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { = - {{\left( {\frac{1}{{n - 2}}} \right)}^2}{t^{\frac{{n - 1}}{{2 - n}}}}\int_{\Gamma (t)} | x{|^{ - \mu }}{u^{ - p}}{\rm{d}}\sigma }\\ { = - {{\left( {\frac{1}{{n - 2}}} \right)}^2}{t^{\frac{{n - 1 - \mu }}{{2 - n}}}}\int_{\Gamma (t)} {{u^{ - p}}} {\rm{d}}\sigma }\\ { \le - {{\left( {\frac{1}{{n - 2}}} \right)}^2}{t^{\frac{{n - 1 - \mu }}{{2 - n}}}}|\Gamma (t){|^{1 + p}}{{\left( {\int_{\Gamma (t)} u {\rm{d}}\sigma } \right)}^{ - p}}}\\ { = - {d_2} \cdot {t^{\frac{{n - 1 - \mu + (n - 1)(1 + p)}}{{2 - n}}}}{{\left( {\int_{\Gamma (t)} u {\rm{d}}\sigma } \right)}^{ - p}}}\\ { = - {d_2} \cdot {t^{\frac{{n - 1 - \mu + (n - 1)(1 + p) + (1 - n)p}}{{2 - n}}}} \cdot } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left( {\int_{\Gamma (t)} u |x{|^{1 - n}}{\rm{d}}\sigma } \right)^{ - p}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = - {d_2} \cdot {t^{\frac{{2(n - 1) - \mu }}{{2 - n}}}}A{(t)^{ - p}} \le - {d_3} \cdot {t^{\frac{{2(n - 1) - \mu + lp}}{{2 - n}}}}, \end{array} $

其中d₁, d₂, d₃是与n, μ, p有关、与t无关的正数值。从而，得到关系式

$ {A(t) \le {d_1}{t^{\frac{{ - l}}{{2 - n}}}},} $

(5)

$ {{A^\prime }(t) = \frac{1}{{n - 2}}\int_{\Gamma (t)} {\frac{{\partial u}}{{\partial \mathit{\boldsymbol{n}}}}} {\rm{d}}\sigma ,} $

(6)

$ {A^{\prime \prime }}(t) \le - {d_2} \cdot {t^{\frac{{2(n - 1) - \mu }}{{2 - n}}}}A{(t)^{ - p}} \le - {d_3} \cdot {t^{\frac{{2(n - 1) - \mu + lp}}{{2 - n}}}}. $

(7)

下面分析当t→∞, A(t), A′(t), A″(t)可能出现的情况。由A″(t) < 0, 可知A′(t)单减：

(Ⅰ)若存在t₁>0, 使得当t>t₁时，A′(t)≤A′(t₁) < 0, 则A(t)单减，又A(t)>0, 则在区间[t₁, +∞), 函数A(t)存在最大值M₂, 0 < M₁ < +∞。对∀t>0, $\frac{{2{M_1}}}{t} \geqslant \frac{{A(t + {t_0}) - A({t_0})}}{t} = A' ({\xi _t}) \geqslant \frac{{ - 2{M_1}}}{t}$, ξ_t∈[t₀, t₀+t]。当t→+∞, 有序列A′(ξ_t)→0, 产生矛盾。因此，这种情况不成立。

(Ⅱ)若∀t>0, 有A′(t)≥0, 则A(t)单增，$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t)=+∞或$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t)=M₂>0。若$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t)=M₂，则∀$\epsilon $>0, 存在t充分大时，有0 < A(2t)-A(t) < $\epsilon $, 从而$ 0<~\frac{A\left( 2t \right)-A(t)}{t}=A' ({{\eta }_{t}})<\frac{\epsilon }{t} $, η_t∈[t, 2t], 则$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A′(η_t)=0, 又由A′(t)单减，也就有$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A′(t)=0。

定义函数F(t)

$ F(t) = \int_{A(R)}^{A(t)} {{s^{ - \gamma }}} {\rm{d}}s = \frac{1}{{\gamma - 1}}\left[ {A{{(R)}^{1 - \gamma }} - A{{(t)}^{1 - \gamma }}} \right], $

其中t∈[R, +∞), γ=$\frac{{\mu - 2}}{l}$-p>1。计算F(t)的一阶导数及二阶导数，有

$ {{F^\prime }(t) = A{{(t)}^{ - \gamma }} \cdot {A^\prime }(t),} $

$ {{F^{\prime \prime }}(t) = A{{(t)}^{ - \gamma }}{A^{\prime \prime }}(t) - \gamma A{{(t)}^{ - \gamma - 1}}{A^\prime }(t).} $

已知A′(t)≥0, 且由式(5)，有

$ {A{{(t)}^{ - p - \gamma }} \ge {{\left( {{d_1}{t^{\frac{{ - l}}{{2 - n}}}}} \right)}^{ - p - \gamma }} = d_1^{ - p - \gamma }{t^{\frac{{l(p + \gamma )}}{{2 - n}}}},} $

$ { - A{{(t)}^{ - p - \gamma }} \le - d_1^{ - p - \gamma }{t^{\frac{{l(p + \gamma )}}{{2 - n}}}}.} $

再结合式(7)和γ=$\frac{{\mu - 2}}{l}$-p, 估计F″(t), 有

$ \begin{array}{l} {F^{\prime \prime }}(t) \le A{(t)^{ - \gamma }}{A^{\prime \prime }}(t)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le - {d_2} \cdot {t^{\frac{{2(n - 1) - \mu }}{{2 - n}}}}A{(t)^{ - p - \gamma }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le - {d_4}{t^{\frac{{2(n - 1) - \mu + (\chi + p)}}{{2 - n}}}} = - {d_4}{t^{ - 2}}, \end{array} $

其中d₄是与n, μ, p有关、与t无关的正数值。从对A(t), A′(t), A″(t)的分析，可知F′(t)≥0, ∀t>0, 且单减，则$\mathop {{\text{lim}}}\limits_{t \to + \infty } $F′(t)=M₃, 0≤M₃ < +∞, 且根据情况(Ⅱ), $\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t)=+∞, 且0≤$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A′(t) < +∞，或者0 < $\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t) < +∞，且$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A′(t)=0, 则$\mathop {{\text{lim}}}\limits_{t \to + \infty } $F(t)=$\mathop {{\text{lim}}}\limits_{t \to + \infty } $A(t)^-γ·A′(t)=0。

考虑函数sF″(s), s∈[R, t], 它的积分满足

$ \begin{array}{l} \int_R^t s {F^{\prime \prime }}(s){\rm{d}}s = t{F^\prime }(t) - R{F^\prime }(R) - F(t) + F(R)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le \int_R^t - {d_4}{s^{ - 1}}{\rm{d}}s = - {d_4}(\ln t - \ln R), \end{array} $

则有

$ - R{F^\prime }(R) - F(t) \le - {d_4}(\ln t - \ln R). $

当t→+∞, 不等式左边极限为有限数, 而右边趋于-∞, 产生矛盾，从而定理成立，即对$\forall 0 < l < \frac{\mu -2}{p+1}$, 方程(2)的C²(Ω\{0})-解u都满足性质

$ \mathop {{\rm{limsup}}}\limits_{x \to 0} \frac{{u(x)}}{{|x{|^{ - l}}}} = + \infty . $