In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in \({W^{1, \Phi}_{loc} (\Omega)}\) to the problem$$\left\{\begin{array}{ll}-\Delta_{\Phi}u=f(x,u)\, {\rm in}\, \Omega,\\u > 0\, {\rm in} \, \Omega, u = 0 \,{\rm on}\, \partial\Omega,\end{array}\right.$$where f has \({\Phi}\)-sublinear growth. In our main results the function f(x, u) may be singular at u = 0 and the nonlinear term \({f(x, t)/t^{\ell-1}, t > 0}\) is strictly decreasing for a suitable \({\ell > 1}\). Under different kind of boundary conditions we prove an improvement for the classical Brézis-Oswald and Díaz-Sáa’s results in Orlicz- Sobolev framework for singular nonlinearities as well. Some results discussed here are news even for Laplacian or p-Laplacian operators.

中文翻译：

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$$ {\ Phi} $$Φ的一类Brézis–Oswald问题-具有非常奇异项的Laplacian算子

在这项工作中，我们考虑了拟线性椭圆问题解的存在性和唯一性，该问题在原点可能是奇异的。此外，我们考虑仅在\（{W ^ {1，\ Phi} _ {loc}（\ Omega）} \）中的子解决方案和超级解决方案的比较原理，以解决问题$$ \ left \ {\ begin {array} { ll}-\ Delta _ {\ Phi} u = f（x，u）\，{\ rm in} \，\ Omega，\\ u> 0 \，{\ rm in} \，\ Omega，u = 0 \ ，{\ rm on} \，\ partial \ Omega，\ end {array} \ right。$$，其中f具有\（{\ Phi} \）-次线性增长。在我们的主要结果的函数˚F（X，Û）可以在单数ù = 0和非线性项\（{F（X，T）/ T ^ {\ ELL-1}，在t> 0} \）是严格减少合适的时间\（{\ ell> 1} \）。在不同的边界条件下，我们还证明了经典Brézis-Oswald和Díaz-Sáa在Orlicz-Sobolev框架中用于奇异非线性的结果的改进。即使对于拉普拉斯算子或p-拉普拉斯算子，此处讨论的某些结果也是新闻。