Nonlinear Differential Equations and Applications (NoDEA) ( IF 1.2 ) Pub Date : 2020-06-18 , DOI: 10.1007/s00030-020-00641-z L. Boccardo , S. Buccheri , G. R. Cirmi
In this paper we study the existence and regularity of solutions to some nonlinear boundary value problems with non coercive drift. The model problem is
$$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(A(x)\nabla u|\nabla u|^{p-2} )=E(x)\nabla u|\nabla u|^{p-2}+f(x), &{} \text {in } \Omega ; \\ u =0, &{} \text {on } \partial \Omega ; \end{array}\right. \end{aligned}$$(1)where \(p>1\), \(\Omega \) is an open bounded subset of \({\mathbb {R}}^N\), A(x) is an elliptic matrix with measurable and bounded entries,\(E\in (L^{N}(\Omega ))^N\) and \(f\in L^{m}(\Omega )\) with \(1<m<\frac{N}{p}\). No further regularity on the coefficients of A(x) is used and no smallness assumption of \(\Vert |E|\Vert _{L^{N}(\Omega )}\) is required. Our strategy is based on the proof of a priori estimates by contradiction.
中文翻译：
具有奇异漂移的非线性椭圆方程的无限能量解的Calderon–Zygmund–Stampacchia理论
本文研究了一些具有非强制性漂移的非线性边值问题的解的存在性和规律性。模型问题是
$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ mathrm {div}（A（x）\ nabla u | \ nabla u | ^ {p-2}）= E（x） \ nabla u | \ nabla u | ^ {p-2} + f（x），＆{} \ text {in} \ Omega; \\ u = 0，＆{} \ text {on} \ partial \ Omega; \ end {array} \ right。\ end {aligned} $$（1）其中\（p> 1 \），\（\ Omega \）是\（{\ mathbb {R}} ^ N \）的一个开放有界子集，A（x）是一个具有可测和有界条目的椭圆矩阵，\ （E \ in（L ^ {N}（\ Omega））^ N \）和\（f \ in L ^ {m}（\ Omega）\）与\（1 <m <\ frac {N} {p } \）。不再使用关于A（x）的系数的正则性，也不需要\（\ Vert | E | \ Vert _ {L ^ {N}（\ Omega}} \）的小假设。我们的策略是基于对矛盾的先验估计的证明。
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