## 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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### References:

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