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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.

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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