Positive solutions for a singular and superlinear \(p\)-Laplacian problem with gradient term (Q2008596)

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.