Existence and boundedness of solutions to singular anisotropic elliptic equations

In this talk, we present new results on the existence and uniform boundedness of solutions for a general class of Dirichlet anisotropic elliptic problems

of the form

where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ $(N\ge 2)$, ${\mathrm{\Delta }}_{\overrightarrow{p}}u=\sum _{j=1}^N{\partial }_j(|{\partial }_{j}u{|}^{p_j-2}{\partial }_{j}u)$ and

${\mathrm{\Phi }}_0(u,\mathrm{\nabla }u)=({\mathfrak{a}}_0+\sum _{j=1}^N{\mathfrak{a}}_j|{\partial }_{j}u{|}^{p_j})|u{|}^{m-2}u$,

with ${\mathfrak{a}}_0>0$,

$m,p_j>1$,   ${\mathfrak{a}}_j\ge 0$ for $1\le j\le N$ and $N/p=\sum _{k=1}^N(1/p_k)>1$. We assume that $f\in L^r(\mathrm{\Omega })$ with $r>N/p$. The feature of this study  is the inclusion of a possibly singular gradient-dependent term $\mathrm{\Psi }(u,\mathrm{\nabla }u)=\sum _{j=1}^N|u{|}^{{\theta }_j-2}u|{\partial }_{j}u{|}^{q_j}$, where ${\theta }_j>0$ and $0\le q_j<p_j$ for $1\le j\le N$.

This is joint work with Barbara Brandolini (Università degli Studi di Palermo).