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**Positive solutions for a singular and superlinear \(p\)-Laplacian problem with gradient term.**
*(English)*
Zbl 1435.35190

The paper studies the problem \( -\Delta_p u+\mu \alpha (x)|\nabla u|^p=a(x)f(u)+\lambda b(x)g(u)\) in \(\Omega \), \( u=0\) on \( \partial \Omega \). Here \( \Omega \subset R^N\) is a bounded domain with smooth boundary, \( \Delta_p u=\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) with \( p\in (1,N)\), \( 0\le q\le p\) and \( \lambda \), \( \mu \) are positive parameters. The functions \( f\) and \( g\) are positive and continuous on the interval \( (0,\infty )\). Under some conditions it is proved that there are positive numbers \( \lambda_*\), \( \lambda^*\) and \( \mu^* \) such that for \( \lambda \in (\lambda_*,\lambda^*) \) and \( \mu \in (0,\mu^*)\) there exists a positive solution \( u\in C^1(\Omega )\cap C(\overline \Omega )\) of the problem.

Reviewer: Dagmar Medková (Praha)

### MSC:

35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |

35J75 | Singular elliptic equations |

35B08 | Entire solutions to PDEs |

35B09 | Positive solutions to PDEs |

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\textit{M. C. Rezende} and \textit{C. A. Santos}, Rocky Mt. J. Math. 49, No. 6, 2029--2046 (2019; Zbl 1435.35190)

### References:

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