A quasilinear problem in two parameters depending on the gradient

TL;DR

This paper establishes the existence of positive solutions for a class of quasilinear PDEs with two parameters and gradient dependence, unifying various approaches and extending to nonlinearities involving the gradient.

Contribution

It generalizes the sub- and super-solution method to handle gradient-dependent nonlinearities in p-Laplacian problems, including the homogeneous case and broader classes of functions.

Findings

Existence results for positive solutions under parameter constraints.

Extension of methods to nonlinearities depending on the gradient.

Application to specific gradient-dependent p-Laplacian problems.

Abstract

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }\Omega\\ u & = & 0 & \text{on }\partial\Omega, \end{array} \right. \] where $\lambda$ and $\beta$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $\omega_{1}(x)$ and $\omega_{2}(x)$ 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 $-\Delta_{p}u=f(x,u,\nabla u)$ in $\Omega$, 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 $-\Delta_{p}u=\lambda\omega(x)u^{q-1}\left( 1+|\nabla u|^{p}\right) $ with Dirichlet boundary conditions in the domain $\Omega$.