Existence and uniqueness of positive even homoclinic solutions for second order differential equations
This paper is concerned with the existence of positive even homoclinic solutions for the $p$-Laplacian equation \begin{equation*} (|u'|^{p-2}u')' - a(t)|u|^{p-2}u+f(t,u)=0,\qquad t\in \mathbb{R}, \end{equation*} where $p\ge 2$ and the functions $a $ and $f$ satisfy some reasonable...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2019-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7451 |
| Summary: | This paper is concerned with the existence of positive even homoclinic solutions for the $p$-Laplacian equation
\begin{equation*}
(|u'|^{p-2}u')' - a(t)|u|^{p-2}u+f(t,u)=0,\qquad t\in \mathbb{R},
\end{equation*}
where $p\ge 2$ and the functions $a $ and $f$ satisfy some reasonable conditions. Using the Mountain-Pass Theorem, we obtain the existence of a positive even homoclinic solution. In case $p=2$, the solution obtained is unique under a condition of monotonicity on the function $u\longmapsto \frac{f(t,u)}{u}$. Some known results in the literature are generalized and significantly improved. |
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| ISSN: | 1417-3875 |