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| Let us consider the $p$-Laplacian type equation
\[ \left\{ \begin{array}{lr}
-\ {\rm div} (A(x,u)\ |\nabla u|^{p-2}\ \nabla u) + \frac1p\ A_t(x,u)\ |\nabla u|^p\ =\ f(x,u) & \hbox{in $\Omega$,}\\
u\ = \ 0 & \hbox{on $\partial\Omega$,}
\end{array}\right.\]
where $\Omega \subset {\mathbb R}^N$ is a bounded domain, $N\ge 2$, $p > 1$, $A : \Omega \times {\mathbb R} \to {\mathbb R}$ is a given function which admits the partial derivative $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $f : \Omega \times {\mathbb R} \to {\mathbb R}$ is asymptotically $p$-linear at infinity.
Under suitable hypotheses both at the origin and at infinity and by using variational tools and index theories we prove some multiplicity results in the non-resonant case when $A(x,\cdot)$ is even and $f(x,\cdot)$ is odd.
Joint works with Giuliana Palmieri and Kanishka Perera. |
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