We study the Dirichlet problem for the quasilinear partial differential equation \(\triangle u(z) = h(z)\cdot f(u(z))\) in the unit disk \({\mathbb {D}}\subset {\mathbb {C}}\) with arbitrary continuous boundary data \(\varphi :\partial {\mathbb {D}}\rightarrow {\mathbb {R}}\). The multiplier \(h:{\mathbb {D}}\rightarrow {\mathbb {R}}\) is assumed to be in the class \(L^p({\mathbb {D}}),\)\(p>1,\) and the continuous function \(f:\mathbb {R}\rightarrow {\mathbb {R}}\) is such that \(f(t)/t\rightarrow 0\) as \(t\rightarrow \infty .\) Applying the potential theory and the Leray–Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class \(W^{2,p}_{\mathrm{loc}}({\mathbb {D}})\). Furthermore, we show that \(u\in C^{1,\alpha }_{\mathrm{loc}}({\mathbb {D}})\) with \(\alpha = (p-2)/p\) if \(p>2\) and, in particular, with arbitrary \(\alpha \in (0,1)\) if the multiplier h is essentially bounded. In the latter case, if in addition \(\varphi \) is Hölder continuous of some order \(\beta \in (0,1)\), then u is Hölder continuous of the same order in \(\overline{{\mathbb {D}}}\). We extend these results to arbitrary smooth (\(C^1\)) domains.

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关于平面上的拟线性Poisson方程

我们研究了单位圆盘\ {{\ mathbb {D}} \ subset中的拟线性偏微分方程\（\ triangle u（z）= h（z）\ cdot f（u（z））\）的Dirichlet问题{\ mathbb {C}} \）和任意连续边界数据\（\ varphi：\ partial {\ mathbb {D}} \ rightarrow {\ mathbb {R}} \）。假定乘数\（h：{\ mathbb {D}} \ rightarrow {\ mathbb {R}} \）在类\（L ^ p（{\ mathbb {D}}），\）\（ p> 1，\）和连续函数\（f：\ mathbb {R} \ rightarrow {\ mathbb {R}} \）使得\（f（t）/ t \ rightarrow 0 \）为\（t \ rightarrow \ infty。\）加电位理论和勒雷- Schauder不的方法，我们证明了连续解的存在性ü在索伯列夫类问题的\（W ^ {2页} _ {\ mathrm {LOC}}（{\ mathbb {d} }）\）。此外，我们证明\（u \ in C ^ {1，\ alpha} _ {\ mathrm {loc}}（{\ mathbb {D}}）\）与\（\ alpha =（p-2）/ p \）如果为\（p> 2 \），并且尤其是如果乘数h是有界的，则使用任意\（\ alpha \ in（0,1）\）。在后一种情况下，如果另外\（\ varphi \）是某个阶\（\ beta \ in（0,1）\）的Hölder连续，则u是相同阶的Hölder连续\（\ overline {{{\ mathbb {D}}} \）。我们将这些结果扩展到任意平滑（\（C ^ 1 \））域。