| Abstract: |
| We discuss the existence of zero, one or two $T$-periodic solutions for the parameter dependent $\phi$-Laplacian equation of the form $(\phi(u`))`+f(u)u`+g(t,u)=s$, where $s$ is a real number, $f$ and $g$ are continuous functions and $g$ is $T$-periodic in the variable $t$. Nowadays, a phenomenon modulated by the parameter $s$ of this kind is called ``Ambrosetti-Prodi type alternative``, with reference to the pioneering work of [Ambrosetti and Prodi, Ann. Mat. Pura Appl., 1972]. Inspired by the results carried out in [E.S. and F. Zanolin, Adv. Nonlinear Stud., 2018], we investigate the periodic problem with $\phi$-Laplacian operator for a wide family of nonlinearities under local coercivity conditions on $g$. As a result, we give a generalization, along with a standardization, of various classical and recent results on parameter-dependent nonlinear equations. This is a joint work with G.~Feltrin (University of Turin, Italy) and F.~Zanolin (University of Udine, Italy). |
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