Positive solutions for a singular and superlinear $p$-Laplacian problem with gradient term

Abstract

We deal with positive solutions for the equation $ -\Delta _p u +\mu \alpha (x)|\nabla u|^q = a(x)f(u) + \lambda b(x)g(u)$ in $ \Omega $, where $\Omega \subset \mathbb {R}^N$ is a smooth bounded domain; $\Delta _p$ is the $p$-Laplacian operator with $1\lt p\lt N$; $0 \leq q \leq p$; $\lambda , \mu >0$ are real parameters; $ f: (0, \infty )\rightarrow [0,\infty )$ may be singular at $0$; $g: (0, \infty )\rightarrow [0,\infty )$ may be superlinear and $\alpha , a, b:{\Omega } \to (0,\infty )$ are suitable functions. The main novelties in this paper are the presence of singular and superlinear nonlinearities linked with the gradient term, which make it impossible to apply standard comparison principles and variational techniques. We overcome these difficulties by improving a technique of regularization-monotonicity of nonlinearities $f$ and $g$, using sub- and supersolution methods and avoiding the direct use of comparison principles to the operator $ -\Delta _p u +\mu \alpha (x)|\nabla u|^q$.