Recent zbMATH articles in MSC 34B40https://www.zbmath.org/atom/cc/34B402021-04-16T16:22:00+00:00WerkzeugExistence of homoclinic orbits for a singular differential equation involving \(p\)-Laplacian.https://www.zbmath.org/1456.340452021-04-16T16:22:00+00:00"Yin, Honghui"https://www.zbmath.org/authors/?q=ai:yin.honghui"Du, Bo"https://www.zbmath.org/authors/?q=ai:du.bo"Yang, Qing"https://www.zbmath.org/authors/?q=ai:yang.qing"Duan, Feng"https://www.zbmath.org/authors/?q=ai:duan.fengIn the present manuscript, the authors are concerned with the existence of {homoclinic} solutions for the following singular ODE
\[
\Big(\Phi_p\big(x'(t)\big)\Big)'+f\big(x'(t)\big) + g\big(x(t)\big) + \frac{h(t)}{1-x(t)} = e(t), \tag{1}
\]
where $\Phi_p(s) = |s|^{p-2}s$ (for some $p > 1$), $f,g,h,e\in C(\mathbb{R};\mathbb{R})$ and, moreover, $h$ is a strictly positive $T$-periodic function.
As usual, a \textit{homoclinic solution} of (1) is a solution $x\in C(\mathbb{R};\mathbb{R})$ satisfying
\[
\text{$x(t)\to\infty$ as $|t|\to\infty$}.
\]
Due to their relevance in several contexts, homoclinic solutions for general differential systems have been studied by many authors and with different techniques (variational methods, critical-point theory, method of lower/upper solutions and fixed-point theorems, etc.); however, since equation (1) is strongly nonlinear, these traditional techniques are no-longer applicable.
Using a new continuation theorem due to Manásevich and Mawhin, the authors obtain the following theorem, which is the main result of the paper.
Theorem 1.
Assume that the following assumptions are satisfied:
\begin{itemize}
\item[{(H.1)}] $f:\mathbb{R}\to\mathbb{R}$ is continuous, bounded and non-negative;
\item[{(H.2)}] $g:\mathbb{R}\to\mathbb{R}$ is strictly monotone increasing and there are positive constants $\sigma$ and $n$ such that
\[
xg(x)\geq \sigma|x|^{n+1}\quad\text{for all $x\in\mathbb{R}$};
\]
\item[{(H.3)}] $\rho_1 := \sup_{t\in\mathbb{R}}|e(t)| < \infty$ and
\[
\rho_2 := \int_{\mathbb{R}}|e(t)|^{1+1/n}\,\mathrm{d} t < \infty.
\]
\end{itemize}
Then, if $\rho_1 > f(0)$ and $h_l/\rho_1 - f(0) < 1$ (with $h_l := \min_{t\in\mathbb{R}}h(t)$), there exists at least one positive homoclinic solution $\omega_0$, further satisfying
\[
|\omega_0'(t)|\to 0\quad\text{as $|t|\to\infty$}.
\]
Thought it is based on the continuation theorem by Manásevich and Mawhin, the proof of Theorem 1 is sophisticated and it requires some preliminary lemmas of independent interest. On the other hand, a couple of examples at the end of the paper show the wide range of applicability of this result.
Reviewer: Stefano Biagi (Milano)