| Abstract: |
| Let us consider the class of asymptotically ``$p (s + 1)$-linear`` $p$-Laplacian problems
\[
\left\{
\begin{array}{ll}
- {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right]
+ s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\
\qquad\qquad\qquad =\ \mu |u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $\Omega$,}\
u = 0 & \hbox{on $\partial\Omega$,}
\end{array}
\right.
\]
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$,
$1 < p < N$, $s > 1/p$,
both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(\Omega)$
and far away from 0, $\mu \in \mathbb{R}$,
and the ``perturbation`` term
$g(x,t)$ grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that
$g(x,t) \approx \nu |t|^{p-2} t$ as $t \to 0$.
Under good hypotheses on $g(x,t)$,
suitable thresholds for the parameters $\mu$ and $\nu$
exist so that
the existence of a nontrivial weak solution of the given problem
is proved if either $\nu$ is large enough with $\mu$ small enough
or $\nu$ is small enough with $\mu$ large enough.
$$ $$
Joint work with Kanishka Perera and Addolorata Salvatore.
$$ $$
Partially supported by MUR PRIN 2022 PNRR Research Project
P2022YFAJH, Linear and Nonlinear PDEs: New Directions and Applications. |
|