Existence results for $p$-Laplace equations with nonlinearities having growth in the function and its gradient

Abstract

In a bounded domain $\mathrm{\Omega }\subset {ℝ}^N$, $N\ge 2$, we study the solvability of the boundary value problem

$$-{\mathrm{\Delta }}_{p}u=H\left(x,u,\nabla u\right)\text{ in}\mathrm{\Omega },u=0\text{ on}\partial \mathrm{\Omega },$$

where $-{\mathrm{\Delta }}_{p}u=-div\left(|\nabla u{|}^{p-2}\nabla u\right)$ is the $p$-Laplace operator with $1<p<N$ and the nonlinearity $H$ satisfies $\left|H\left(x,u,\nabla u\right)\right|\le a|\nabla u{|}^q+b|u{|}^r+f\left(x\right)$ with $a,b\ge 0$ and $q,r>0$. Assuming $f\in L^m\left(\mathrm{\Omega }\right)$ for a suitable exponent $m>1$, we prove the existence of solutions for different sets of values $\left(q,r\right)$. To this end, we apply the classical Schauder fixed point theorem, relying on some well-known uniqueness, regularity estimates and stability results for renormalized and weak solutions. In the last section we make some examples to illustrate the applicability of our results to a large class of equations.