For an open, bounded domain \(\Omega \) in \(\mathbb {R}^N\) which is strictly convex with smooth boundary, we show that there exists a \(\Lambda >0\) such that for \(0<\lambda <{\Lambda } \), the quasilinear singular problem$$\begin{aligned} \begin{aligned} -\Delta _pu&= \lambda u^{-\delta }+u^q\,\,\text { in }\,\,\Omega \\ u&= 0\,\,\text { on }\,\,\partial \Omega ;\, \,\,u>0\,\,\text { in }\,\,\Omega \end{aligned} \end{aligned}$$admits at least two distinct solutions u and v in \(W^{1,p}_{loc}(\Omega )\cap L^{\infty }(\Omega )\) provided \(\delta \ge 1\), \(\frac{2N+2}{N+2}<p<N\) and \(p-1<q<\frac{Np}{N-p}-1\).

中文翻译：

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一类具有奇异非线性的拟线性方程的解的多重性

对于\（\ mathbb {R} ^ N \）中的一个开放的有界域\（\ Omega \），它是严格凸且具有平滑边界的，我们证明存在一个\（\ Lambda> 0 \）使得对于\ （0 <\ lambda <{\ Lambda} \），拟线性奇异问题$$ \ begin {aligned} \ begin {aligned}-\ Delta _pu＆= \ lambda u ^ {-\ delta} + u ^ q \，\ ，\ text {in} \，\，\ Omega \\ u＆= 0 \，\，\ text {on} \，\，\ partial \ Omega; \，\，\，u> 0 \，\，\ text {在} \，\，\欧米茄\ {端对齐} \ {端对齐} $$坦承至少两种不同的解决方案ù和v在\（W ^ {1，p} _ {LOC}（\欧米茄）\帽L ^ {\ infty}（\ Omega} \）提供\（\ delta \ ge 1 \），\（\ frac {2N + 2} {N + 2} <p <N \）和\（p-1 <q <\ frac {Np} {Np} -1 \）。