I will present some existence results for a class of nonlinear elliptic problems whose prototype is
\[\left\{
\begin{array}{ll}
-\text{div}(|\nabla u|^{p-2} \nabla u)= \beta |\nabla u|^q + c|u|^{p-2} u + f &\text{in }\Omega,\\
u=0&\text{on }\partial\Omega,
\end{array}\right.
\]
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\), \(N\geq2\), \(1\leq p\leq N\), \(p-1\leq q\leq p\), \(\beta\) is a positive
constant. Moreover the coefficient \(c\) and the datum \(f\) are measurable functions with suitable
summability depending on the interval of the values of \(q\) and they also satisfy a smallness
condition on their norms.
The existence results are contained in a joint paper with A. Alvino and V. Ferone, when
the zero order term does not appear, or in some joint papers with A. Alvino, M.F. Betta
and R. Volpicelli, where the case of the complete elliptic operators has been studied.
