Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Wed, 21 Apr 2021 14:14:49 GMT2021-04-21T14:14:49Z101001Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutionshttp://hdl.handle.net/11368/2504937Titolo: Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions
Abstract: We develop a lower and upper solutions method for the periodic problem associated with the capillarity equation
\begin{equation*}
-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u)
\end{equation*}
in the space of bounded variation functions. We get the existence of periodic
solutions both in the case where the lower solution
$\alpha$ and the upper solution
$\beta$ satisfy $\alpha \le \beta$, and in the case where
$\alpha \not\le \beta$.
In the former case we also prove regularity and order stability of solutions.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11368/25049372012-01-01T00:00:00ZMultiple bounded variation solutions of a periodically perturbed sine-curvature equationhttp://hdl.handle.net/11368/2309122.3Titolo: Multiple bounded variation solutions of a periodically perturbed sine-curvature equation
Abstract: We prove the existence of at least two $T$-periodic solutions, not differing from each other by an integer multiple of $2\pi$, of the sine-curvature equation$$-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = A \sin u + h(t).$$We assume that $A\in\RR$and $h\in L^1_{\rm loc}(\RR)$ is a $T$-periodic function such that $\int_0^T h \, dt=0$ and, e.g., $\|h\|_{L^\infty} < 4/T$. Our approach is variational and makes use of basic results of non-smooth critical point theory in the space of bounded variation functions.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11368/2309122.32011-01-01T00:00:00ZAn elliptic problem with arbitrarily small positive solutionshttp://hdl.handle.net/11368/2311256Titolo: An elliptic problem with arbitrarily small positive solutions
Abstract: We show that for each $\lambda > 0$, the problem
$-\Delta_p u = \lambda f(u)$ in $Omega$,
$u = 0$ on $\partial \Omega$
has a sequence of positive solutions $(u_n)_n$
with $\max_{\bar\Omega} u_n$ decreasing to zero.
We assume that $\displaystyle{\liminf_{s\to0^+}\frac{F(s)}{s^p} = 0}$
and that
$\displaystyle{\limsup_{s\to 0^+}\frac{F(s)}{s^p} = +\infty}$,
where $F'=f$. We stress that no condition on the sign of $f$ is imposed.
Sat, 01 Jan 2000 00:00:00 GMThttp://hdl.handle.net/11368/23112562000-01-01T00:00:00ZOn the Ambrosetti-Prodi problem for first order scalar periodic ODEshttp://hdl.handle.net/11368/1720966Titolo: On the Ambrosetti-Prodi problem for first order scalar periodic ODEs
Abstract: We survey some classical and recent results about the Ambrosetti-Prodi
problem for the scalar first order periodic ordinary differential
equation
$x'=f(t,x)$. This problem plays a role in describing
the evolution of single species populations subject to periodic
fluctuations and periodic harvesting, as well as in studying some
special cases of Hilbert sixteenth problem.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11368/17209662005-01-01T00:00:00ZPeriod two implies any period for a class of differential inclusionshttp://hdl.handle.net/11368/2337049Titolo: Period two implies any period for a class of differential inclusions
Abstract: We produce a detailed proof of a result stated in "F. Obersnel and P. Omari, Period two implies chaos for a class of ODEs, Proc. Amer. Math. Soc, 135 (2007), 2055-2058" concerning scalar time-periodic first order differential inclusions. Such a result shows that the existence of just one subharmonic implies the
existence of large sets of subharmonics of all given orders.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11368/23370492006-01-01T00:00:00ZOn the periodic Ambrosettiâ€“Prodi problem for a class of ODEs with nonlinearities indefinite in signhttp://hdl.handle.net/11368/2969071Titolo: On the periodic Ambrosettiâ€“Prodi problem for a class of ODEs with nonlinearities indefinite in sign
Abstract: We prove a result of Ambrosetti-Prodi type for the scalar periodic ODE $x'=f(t,x)-s$, where, seemingly for the first time in the literature, $f(cdot,x) $ is allowed to have indefinite sign as $|x| o+infty$. Our result requires that $f$ satisfies a one-sided growth control; in case such a control fails, non-existence occurs for large $s>0$, although multiplicity of solutions can still be detected provided $f(cdot,0)=0$ and $s>0$ is small enough.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29690712020-01-01T00:00:00ZSubharmonic solutions of the prescribed curvature equationhttp://hdl.handle.net/11368/2830905Titolo: Subharmonic solutions of the prescribed curvature equation
Abstract: We study the existence of subharmonic solutions of the prescribed curvature equation
\begin{equation*}
-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u).
\end{equation*}
According to the behaviour at zero, or at infinity, of the prescribed curvature $f$, we prove the existence of arbitrarily small classical subharmonic solutions, or bounded variation subharmonic solutions with arbitrarily large oscillations.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11368/28309052016-01-01T00:00:00ZPositive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equationhttp://hdl.handle.net/11368/2507945Titolo: Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation
Abstract: We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation
\begin{equation*}
-\Big( u'/{ \sqrt{1-{u'}^2}}\Big)'
= f(t,u).
\end{equation*}
Depending on the behaviour of $f=f(t,s)$ near $s=0$, we prove the existence of either
one, or two, or three, or infinitely many positive solutions. In general,
the positivity of $f$ is not required.
All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11368/25079452012-01-01T00:00:00ZPositive solutions of the Dirichlet problem for the prescribed mean curvature equationhttp://hdl.handle.net/11368/2300181Titolo: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation
Abstract: We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem\begin{equation*}-{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = \lambda f(x,u)\mbox{\, in $\Omega$},\qquadu=0 \mbox{\, on $\partial \Omega$},\end{equation*}in a general bounded domain $\Omega\subset\RR^N$, depending on the behaviour at zero or at infinity of $f(x,s)$, or of its potential $F(x,s)=\int_0^s f(x,t)\,dt$. Our main effort here is to describe, in a way as exhaustive as possible, all configurations of the limits of $F(x,s)/s^2$ at zero and of $F(x,s)/s$ at infinity, which yield the existence of one, two, three or infinitely many positive solutions. Either strong, or weak, or bounded variation solutions are considered. Our approach is variational and combines critical point theory, the lower and upper solutions method and elliptic regularization.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/11368/23001812010-01-01T00:00:00ZExistence and multiplicity results for the prescribed mean curvature equation via lower and upper solutionshttp://hdl.handle.net/11368/2280921Titolo: Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions
Abstract: We discuss existence and multiplicity of bounded variation solutions of the mixed problem for the prescribed mean curvature equation$$-{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = f(x,u) \hbox{\, in $\Omega$},\quadu=0 \hbox{\, on $\Gamma_{D}$}, \quad \partial u / \partial \nu =0 \hbox{\, on $ \Gamma_{N}$}, $$where $\Gamma_{D} $ is an open subset of $\partial \Omega$ and $\Gamma_{N}=\partial \Omega\setminus \Gamma_{D}$. Our approach is based on variational techniques and a lower and upper solutions method specially developed for this problem.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11368/22809212009-01-01T00:00:00Z