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Singular quasilinear elliptic problems with changing sign datum: existence and homogenization
Revista Matemática Complutense ( IF 0.8 ) Pub Date : 2019-07-01 , DOI: 10.1007/s13163-019-00313-2
José Carmona , Salvador López-Martínez , Pedro J. Martínez-Aparicio

We study singular quasilinear elliptic equations whose model is$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\,\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) &{} \quad \hbox {in}\,\, \Omega , \\ u=0 &{}\quad \hbox {on} \,\, \partial \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\) (\(N\ge 3\)), \(\lambda \in {\mathbb {R}}\), \(1<q< 2\), \(0\le \mu \in L^\infty (\Omega )\) and the datum \(f\in L^p(\Omega )\), for some \(p>\frac{N}{2}\), may change sign. We prove existence of solution and we deal with the homogenization problem posed in a sequence of domains \(\Omega ^\varepsilon \) obtained by removing many small holes from a fixed domain \(\Omega \).

中文翻译:

改变符号原点的奇异拟线性椭圆问题:存在与均质化

我们研究模型为$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ displaystyle-\,\ Delta u = \ lambda u + \ mu(x)\ frac { | \ nabla u | ^ q} {| u | ^ {q-1}} + f(x)&{} \ quad \ hbox {in} \,\,\ Omega,\\ u = 0&{} \ Quad \ hbox {on} \,\,\ partial \ Omega,\ end {array} \ right。} \ end {aligned} $$其中\(\ Omega \)\({\ mathbb {R}} ^ N \)\(N \ ge 3 \)),\(\ lambda \在{\ mathbb {R}} \)中\(1 <q <2 \)\(0 \ le \ mu \ in L ^ \ infty(\ Omega)\)和基准\(f \ in L ^ p(\ Omega)\),对于某些\(p> \ frac {N} {2} \),可能会更改符号。我们证明了解的存在,并且处理了通过从固定域\(\ Omega \)去除许多小孔而获得 的一系列域\(\ Omega ^ \ varepsilon \)构成的均质化问题。
更新日期:2019-07-01
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