Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Mon, 25 Jan 2021 11:39:39 GMT2021-01-25T11:39:39Z101001Positive 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/11368/25079452012-01-01T00:00:00ZOn 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/11368/29690712020-01-01T00:00:00ZGiovanni Torelli, un matematico appassionato: il profilo scientifico, in brevehttp://hdl.handle.net/11368/2958737Titolo: Giovanni Torelli, un matematico appassionato: il profilo scientifico, in breve
Abstract: The figure of the Italian mathematician Giovanni Torelli, who spent his scientific life in Trieste between the late fifties and the late 1980s, is illustrated with reference to his activities in the field of mathematics research. The main topics of his scientific production, as well as his publications, are synthetically described.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11368/29587372019-01-01T00:00:00ZRegular versus singular solutions in a quasilinear indefinite problem with an asymptotically linear potentialhttp://hdl.handle.net/11368/2961235Titolo: Regular versus singular solutions in a quasilinear indefinite problem with an asymptotically linear potential
Abstract: The aim of this paper is analyzing the positive solutions of the quasilinear
problem
egin{equation*}
label{P}
-(u'/sqrt{1+(u')^2})' = lambda a(x) f(u) ; ; ext{in } (0,1),
u'(0)=0,;u'(1)=0,
end{equation*}
where $lambdain R$ is a parameter, $ain L^infty(0,1)$ changes sign once in $(0,1)$
and satisfies $int_0^1a(x),dx<0$, and $f in mc{C}^1(R)$ is positive and increasing in $(0,+infty)$ with a potential, $F(s)=int_0^{s}f(t),dt$, quadratic at zero and linear at $+infty$.
The main result of this paper establishes that
this problem possesses a component of positive bounded variation solutions, $mathscr{C}_{l_0}^+$, bifurcating from $(l,0)$ at some $l_0>0$ and from $(l,infty)$ at some $l_infty>0$.
It also establishes that $mathscr{C}_{l_0}^+$ consists of regular solutions, if, and only if,
centerline{
$
int_0^z left( int_x^z a(t),dt
ight)^{-rac{1}{2}}dx =+infty, quad hbox{or}quad
int_z^1 left( int_x^z a(t),dt
ight)^{-rac{1}{2}}dx =+infty.
$}
Equivalently, the small positive regular solutions of $mathscr{C}_{l_0}^+$ become singular
as they are sufficiently large if, and only if,
centerline{
$
left( int_x^z a(t),dt
ight)^{-rac{1}{2}}in L^1(0,z) quad ext{and} quad
left( int_x^z a(t),dt
ight)^{-rac{1}{2}}in L^1(z,1).
$}
This is achieved by providing a very sharp description of the asymptotic profile, as $l ol_infty$, of the solutions.
According to the mutual positions of $l_0$ and $l_infty$, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29612352020-01-01T00:00:00ZPositive solutions of indefinite logistic growth models with flux-saturated diffusionhttp://hdl.handle.net/11368/2963071Titolo: Positive solutions of indefinite logistic growth models with flux-saturated diffusion
Abstract: This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator
- div (del u/root 1+vertical bar del u vertical bar(2)) - lambda a(x)f(u) in Omega, u - 0 on partial derivative Omega,
with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Omega is a bounded domain in R-N with a regular boundary partial derivative Omega, lambda > 0 represents a diffusivity parameter, a is a continuous weight which may change sign in Omega, and f: [0, L] -> R, with L > 0 a given constant, is a continuous function satisfying f(0) = f (L) = 0 and f (s) > 0 for every s is an element of [0, L]. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying lambda. Typically, the solutions we find are regular as long as lambda is small, while as a consequence of the saturation of the flux they may develop singularities when A becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f (s) = s(L - s) and a 1, having no similarity with the case of linear diffusion based on the Fick-Fourier's law
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29630712020-01-01T00:00:00ZStability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutionshttp://hdl.handle.net/11368/1697215Titolo: Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/11368/16972152003-01-01T00:00:00ZCharacterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operatorhttp://hdl.handle.net/11368/2955597Titolo: Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator
Abstract: The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem egin{equation*} label{P} -(u'/sqrt{1+(u')^2})' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, end{equation*} where $lambdain R$ is a parameter, $ain L^infty(0,1)$ changes sign once in $(0,1)$ at the point $zin(0,1)$, and $f in mc{C}(R)cap mc{C}^1[0, +infty)$ is positive and increasing in $(0,+infty)$ with a potential, $ int_0^{s}f(t),dt$, superlinear at $+infty$. oindent In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as $l o 0^+$, we can characterize the existence of singular bounded variation solutions solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition medskip egin{center} $left( int_x^z a(t),dt ight)^{- rac{1}{2}}in L^1(0,z) $quad and quad $ left( int_x^z a(t),dt ight)^{- rac{1}{2}}in L^1(z,1).$ end{center} medskip No previous result of this nature is known in the context of the theory of superlinear indefinite problems.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29555972020-01-01T00:00:00ZPositive solutions of superlinear indefinite prescribed mean curvature problemshttp://hdl.handle.net/11368/2955697Titolo: Positive solutions of superlinear indefinite prescribed mean curvature problems
Abstract: This paper analyzes the superlinear indefinite prescribed mean curvature problem [ -mathrm{div}left({ abla u}/{sqrt{1+| abla u|^2}} ight)=lambda a(x)h(u) quad ext{in }Omega,qquad u=0 quad ext{on } partialOmega, ] where $Omega$ is a bounded domain in $mathbb{R}^N$ with a regular boundary $partial Omega$, $hin C^0(mathbb{R}) $ satisfies $h(s) sim s^{p}$, as $s o0^+$, $p>1$ being an exponent with $p< rac{N+2}{N-2}$ if $Ngeq 3$, $lambda> 0$ represents a parameter, and $ain C^0(overline Omega) $ is a sign-changing function. The main result establishes the existence of positive regular solutions when $lambda$ is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for $lambda$ small is further discussed assuming that $h$ satisfies $h(s) sim s^{q}$ as $s o +infty$, $q>0$ being such that $q< rac{1}{N-1}$ if $Ngeq 2$; thus, in dimension $Nge 2$, the function $h$ is not superlinear at $+infty$, although its potential $H(s) = int_0^sh(t) mathrm{d}t$ is. Imposing such different degrees of homogeneity of $h$ at $0$ and at $+infty$ is dictated by the specific features of the mean curvature operator.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29556972020-01-01T00:00:00ZExistence, 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/11368/25049372012-01-01T00:00:00ZClassical and non-classical positive solutions of a prescribed curvature equation with singularitieshttp://hdl.handle.net/11368/2262190Titolo: Classical and non-classical positive solutions of a prescribed curvature equation with singularities
Abstract: We investigate the existence of positive solutions of a prescribed curvature equation with singular curvature. Our approach relies on the method of lower- and upper-solutions and truncation arguments.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/11368/22621902007-01-01T00:00:00Z