A quasilinear problem in two parameters depending on the gradient

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -Δ_{p}u & = & λω_{1}(x)\left\vert u\right\vert ^{q-2}% u+βω_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }Ω\\ u & = & 0 & \text{on }\partialΩ, \end{array} \right. \] where $λ$ and $β$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $ω_{1}(x)$ and $ω_{2}(x)$ are nonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled by making $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based on the sub- and super-solution method. The core of the proof of this problem is then generalized to the Dirichlet problem $-Δ_{p}u=f(x,u,\nabla u)$ in $Ω$, where $f$ is a nonnegative, continuous function satisfying simple, geometrical hypotheses. This approach might be considered as a unification of arguments dispersed in various papers, with the advantage of handling also nonlinearities that depend on the gradient, even in the $p$-growth case. It is then applied to the problem $-Δ_{p}u=λω(x)u^{q-1}\left( 1+|\nabla u|^{p}\right)$ with Dirichlet boundary conditions in the domain $Ω$.

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