论文标题
各向异性椭圆方程，具有梯度依赖的下订单项和$ l^1 $数据
Anisotropic elliptic equations with gradient-dependent lower order terms and $L^1$ data
论文作者
论文摘要
我们证明了一般类别各向异性椭圆问题（例如$ \ Mathcal au+φ（x，x，u，u，\ nabla u）= \ mathfrak {b} u+f $ in $ω$，$ω$是$ \ mathbb r^n $ uny $ uncior的$ uniany us uncop，我们证明了较弱的解决方案。主要部分是差异非线性各向异性操作员$ \ Mathcal a $，其原型是$ \ Mathcal a u = - \ sum_ { n $和$ \ sum_ {j = 1}^n（1/p_j）> 1 $。作为本文的新颖性，我们的较低术语涉及一类新的运营商$ \ mathfrak b $，以至于$ \ mathcal {a} - \ mathfrak {b} $有限，强制性和伪单酮从$ w_0^{1，\ offrightarrow {p {p}} $ cription nordift and and and a a a a a a a a a a a a a a a and and and an $φ$具有梯度的“各向异性自然增长”和良好的标志条件。
We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal Au+Φ(x,u,\nabla u)=\mathfrak{B}u+f$ in $Ω$, where $Ω$ is a bounded open subset of $\mathbb R^N$ and $f\in L^1(Ω)$ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $\mathcal A$, the prototype of which is $\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u)$ with $p_j>1$ for all $1\leq j\leq N$ and $\sum_{j=1}^N (1/p_j)>1$. As a novelty in this paper, our lower order terms involve a new class of operators $\mathfrak B$ such that $\mathcal{A}-\mathfrak{B}$ is bounded, coercive and pseudo-monotone from $W_0^{1,\overrightarrow{p}}(Ω)$ into its dual, as well as a gradient-dependent nonlinearity $Φ$ with an "anisotropic natural growth" in the gradient and a good sign condition.
