| Abstract: |
| In this talk we consider boundary value problems on the of the following form
$$\begin{cases}
\displaystyle \Big(\Phi\big(a(t,x(t))\,x`(t)\big)\Big)` = f(t,x(t),x`(t)) \quad
\text{a.e.\,on $\mathbb{R}$} \[0.15cm]
\,\,x(-\infty) = \nu_1,\,\,x(\infty) = \nu_2,
\end{cases}
$$
where $\Phi:\mathbb{R}\to\mathbb{R}$ is a strictly increasing homeomorphism extending the one-dimensional $p$-Laplacian, $a\in C(\Lambda\times\mathbb{R},\mathbb{R})$ is non-negative which can vanish on a set of zero Lebesgue measure, and $f$ is a Carathe\`{o}dory function on $\Lambda\times\R^2$. Under very ge\-ne\-ral assumptions on the functions $a$ and $f$, including an appropriate version of the well-known Nagumo-Wintner growth condition, we prove the existence of at least one solution of the above problem in a suitable Sobolev space.
Our approach combines a fixed-point technique with the method of lower/upper solutions, and is powerful enough to allows to treat both the cases
$$
\begin{array}{c}
\displaystyle f(t,x,y)\thickapprox \frac{1}{|t|^\gamma}\,\,
(\text{for some $\gamma > 1$})\quad\text{and}
\quad
f(t,x,y)\thickapprox \frac{1}{|t|} \[0.3cm]
(\text{\small{as $|t|\to\infty$}})
\end{array}$$ |
|