论文标题
具有梯度依赖性下阶项的单数各向异性椭圆方程
Singular anisotropic elliptic equations with gradient-dependent lower order terms
论文作者
论文摘要
我们证明了在有限的开放子集的$ \ mathbb r^n $中，使用$ n \ ge 2 $的有界开放子集$ω$在有限的开放子集$ω$中存在解决方案，但要遵循均质边界条件：\ begin {equination {equination} \ label} \ label {eq0}} {eq0}} \ left \ lew \ webt \ webt^aray { φ（u，\ nabla u）=ψ（u，\ nabla u）+ \ mathfrak {b} u \ quad＆\ mbox {in}ω，\\ u = 0＆\ mbox {on} \partialΩ。 \ end {array} \ right。在这里$ w_0^{1，\ costrightArrow {p}}（ω）$ to $ w^{ - 1，\ costrightArrow {p}'}'}（ω）$满足适合但一般的结构假设。 $φ$和$ψ$是梯度依赖性的非线性，其模型如下：\ begin {equination*} \ label {phi}φ（u，\ nabla u）：= \ left（\ sum_ = 1} u |^{p_j} +1 \ right）| u |^{m-2} u，\quadψ（u，\ nabla u）：= \ frac {1} {u} {u} \ sum_ {j = 1} \ end {equation*}我们在整个过程中都假设，每$ 1 \ leq j \ leq n $，\ begin {equination*} \ label {sass} \ mathfrak {a} _j \ geq 0，\Quadθ_j> 0 \ mbox {and} \ quad p <n，\ end {equation*}，我们区分两个情况：1）对于每个$ 1 \ leq j \ leq n $，我们都有$θ_j\ geq 1 $; 2）存在$ 1 \ leq j \ leq n $，这样$θ_j<1 $。在最后的情况下，我们寻找\ eqref {eq0}的非负解决方案。
We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $Ω$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{ \begin{array}{ll} \mathcal A u+ Φ(u,\nabla u)=Ψ(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } Ω,\\ u=0 & \mbox{on } \partialΩ. \end{array} \right. \end{equation} Here $ \mathcal A u=-\sum_{j=1}^N |\partial_j u|^{p_j-2}\partial_j u$ is the anisotropic $\overrightarrow{p}$-Laplace operator, while $\mathfrak B$ is an operator from $W_0^{1,\overrightarrow{p}}(Ω)$ into $W^{-1,\overrightarrow{p}'}(Ω)$ satisfying suitable, but general, structural assumptions. $Φ$ and $Ψ$ are gradient-dependent nonlinearities whose models are the following: \begin{equation*} \label{phi}Φ(u,\nabla u):=\left(\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}+1\right)|u|^{m-2}u, \quad Ψ(u,\nabla u):=\frac{1}{u}\sum_{j=1}^N |u|^{θ_j} |\partial_j u|^{q_j}. \end{equation*} We suppose throughout that, for every $1\leq j\leq N$, \begin{equation*}\label{ass} \mathfrak{a}_j\geq 0, \quad θ_j>0, \quad 0\leq q_j<p_j, \quad 1<p_j,m\quad \mbox{and}\quad p<N, \end{equation*} and we distinguish two cases: 1) for every $1\leq j\leq N$, we have $θ_j\geq 1$; 2) there exists $1\leq j\leq N$ such that $θ_j<1$. In this last situation, we look for non-negative solutions of \eqref{eq0}.
