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arXiv 1409.4013math.AP
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具有次自然增长项的拟线性椭圆型方程的有限能量解

Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

Cao Tien Dat, Igor E. Verbitsky

备注：
19 pages, Calc. Var. Partial Differential Equations (2014)
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摘要：

研究了$$ -Δ_pu=σ\, u^q \quad \text{in } \mathbb{R}^n,$$型拟线性椭圆型方程的有限能量解，其中$Δ_p$是$p$ -Laplacian，$p>1$，$σ$是$\mathbb{R}^n$上的非负函数（或测度），在$0<q < p-1$（低于“自然增长”率$q=p-1$）的情况下.给出了在齐次Sobolev空间$L_0^{1,p}(\mathbb{R}^n)$中$σ$存在解的一个显式充要条件，并证明了其唯一性.在我们的主要工具是积分不等式密切相关的这个问题，和沃尔夫潜在的估计用于获得尖锐的解决方案的界限。更一般的拟线性方程的$\mathcal{A}$ -拉普拉斯算子$ \text{div} \mathcal{A}(x,\nabla \cdot)$代替$Δ_p$被认为是以及。

英文摘要：

We study finite energy solutions to quasilinear elliptic equations of the type $$ -Δ_pu=σ\, u^q \quad \text{in } \mathbb{R}^n,$$ where $Δ_p$ is the $p$-Laplacian, $p>1$, and $σ$ is a nonnegative function (or measure) on $\mathbb{R}^n$, in the case $0<q < p-1$ ( below the "natural growth" rate $q=p-1$ ). We give an explicit necessary and sufficient condition on $σ$ which ensures that there exists a solution $u$ in the homogeneous Sobolev space $L_0^{1,p}(\mathbb{R}^n)$, and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the $\mathcal{A}$-Laplacian $ \text{div} \mathcal{A}(x,\nabla \cdot)$ in place of $Δ_p$ are considered as well.

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