In this paper we study local regularity properties of weak solutions to a class of nonlinear noncoercive elliptic Dirichlet problems with \(L^1\) datum. The model example is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p(w)+b(x)|Dw|^{p-1}=f(x)&{}\text {in }\Omega ,\\ w=0&{}\text {on }\partial \Omega . \end{array}\right. } \end{aligned}$$

Here \(\Omega \subset {\mathbb {R}}^N\) is a bounded open subset, \(N>1\), \(-\Delta _p\) is the well known p-Laplace operator, \(1<p<N\), b is a function in the Lorentz space \(L^{N,1}(\Omega )\) and f is a function in \(L^1(\Omega )\). We also investigate similar issues for a lower order perturbation of these problems.

中文翻译：

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具有低阶项的非线性椭圆Dirichlet问题的局部正则性结果

在本文中，我们研究了一类具有\（L ^ 1 \）基准的非线性非矫顽椭圆Dirichlet问题的弱解的局部正则性质。模型示例是

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\ Delta _p（w）+ b（x）| Dw | ^ {p-1} = f（x）＆{} \文本{in} \ Omega，\\ w = 0＆{} \文本{on} \ partial \ Omega。\ end {array} \ right。} \ end {aligned} $$

这里\（\ Omega \ subset {\ mathbb {R}} ^ N \）是有界的开放子集，\（N> 1 \），\（-\ Delta _p \）是众所周知的p -Laplace运算符，\ （1 <p <N \），b是洛伦兹空间\（L ^ {N，1}（\ Omega）\）中的函数，f是\（L ^ 1（\ Omega）\）中的函数。我们还研究了类似的问题，以便对这些问题进行低阶扰动。