| Abstract: |
| Our talk deals with the following nonlinear Dirichlet problem associated to the model equation
$$
-\operatorname{div}\left(a(x) \vert\nabla v\vert^{p-2}\nabla v\right)-\operatorname{div}\left(\vert v\vert^{q(r-1)+1} \vert\nabla v\vert ^{q-2}\nabla v\right)=f\quad \text{in}\; \Omega,
$$
where $\Omega$ is a bounded open subset of $\mathbb{R}^N, N>2,$ $ 1 < q < p \frac{1}{q'}$,
$f$ is a function with poor summability and the function $a(x)$ is a measurable function such that
$$
\alpha \leq a(x) \leq \beta, \quad \text{a.e. } x \in \Omega
$$
with $ \alpha, \beta>0.$
Previous result showed that, even for $f \in L^1\log L^1(\Omega)$, the presence of the degenerate term $-div(\vert v\vert^{q(r-1)+1}|\nabla v|^{q-2}\nabla v)$ yields a strong regularizing effect, ensuring existence and improved regularity of distributional solutions. Building on that, we extended these results to the case where the datum $f \in L^m(\Omega)$, with $m>1$, including the regime $ 1 < m < (p^*)'$. Furthermore, we addressed a new borderline case, obtaining existence result in the space $W^{1,1}_0(\Omega)$. These findings highlight the crucial role played by degenerate gradient terms in overcoming the lack of regularity of the data. |
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