In this paper we focus on the following problem with nonlinear convection term

$$\begin{aligned} {\left\{ \begin{array}{ll} -\,\mathrm{div}(M(x)\nabla u)= -\,\mathrm{div}(u|u|^{\theta -1}E(x))+f(x) \qquad &{} \text{ in } \Omega \,,\\ u (x) = 0 &{} \text{ on } \partial \Omega \,, \end{array}\right. } \end{aligned}$$

where \(\Omega \) is an open bounded domain of \({\mathbb {R}}^N\), with \(N\ge 3\), M(x) is a uniform elliptic matrix with measurable entries, \(\theta \in (0,1)\), \(E(x)\in (L^{r}(\Omega ))^N\), with \(r\in (2,N)\), and \(f(x)\in L^{m}(\Omega )\), with \(m>1\). In particular we study how the relation between the parameters \(\theta \) and r affects existence and summability of solutions.

中文翻译：

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两个线性Dirichlet问题之间的非线性同伦

在本文中，我们关注非线性对流项的以下问题

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\，\ mathrm {div}（M（x）\ nabla u）=-\，\ mathrm {div}（u | u | ^ {\ theta -1} E（x））+ f（x）\ qquad＆{} \ text {in} \ Omega \ ,, \\ u（x）= 0＆{} \ text {on} \ \ Omega \，\ end {array} \ right。} \ end {aligned} $$

其中\（\ Omega \）是\（{\ mathbb {R}} ^ N \）的开放边界域，其中\（N \ ge 3 \），M（x）是具有可测项的均匀椭圆矩阵，\（\ theta \ in（0,1）\），\（E（x）\ in（L ^ {r}（\ Omega））^ N \），其中\（r \ in（2，N）\ ）和\（f ^（x）\ in L ^ {m}（\ Omega）\），以及\（m> 1 \）。特别是，我们研究了参数\（\ theta \）和r之间的关系如何影响解的存在性和可积性。