1,720,989 research outputs found
Unbounded Solutions to Systems of Differential Equations at Resonance
Full text linkWe deal with a weakly coupled system of ODEs of the type xj′′+nj2xj+hj(x1,...,xd)=pj(t),j=1,...,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 π-periodic, nj∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h1, ... , hd are assumed
Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions
Full text linkWe analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of RN, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs
Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight
No full textWe study the problem of the existence and multiplicity of positive
periodic solutions to the scalar ODE
u '' + lambda a(t)g(u) = 0, lambda > 0,
where g(x) is a positive function on R(+), superlinear at zero and
sublinear at infinity, and a(t) is a T-periodic and sign indefinite
weight with negative mean value. We first show the nonexistence of
solutions for some classes of nonlinearities g(x) when lambda is small.
Then, using critical point theory, we prove the existence of at least
two positive T-periodic solutions for lambda large. Some examples are
also provided
Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
Full text linkWe study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations
Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions
Full text linkBoundary value problems of Sturm-Liouville and periodic type for the second order nonlinear ODE u′′ + λ f (t, u) = 0 are considered. Multiplicity results are obtained, for λ positive
and large, under suitable growth restrictions on f (t, u) of superlinear type at u = 0 and of sublinear type at u = ∞. Only one-sided growth conditions are required. Applications are given to the equation u′′ + λq(t)f(u) = 0, allowing also a weight function q(t) of nonconstant
sign
Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems
Full text linkThe paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type (Formula presented.) with d=2 or d=3, bifurcating, for ε small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that U is T-periodic in time) and the case of the fixed-energy problem (assuming that U is independent of time) are considered
Periodic solutions to a forced kepler problem in the plane
Full text linkGiven a smooth function U(t, x), T-periodic in the first variable and satisfying U(t, x) = O(vertical bar x vertical bar(alpha)) for some alpha is an element of (0, 2) as vertical bar x vertical bar -> infinity, we prove that the forced Kepler problem(sic) = -x/vertical bar x vertical bar(3) + del U-x(t, x), x is an element of R-2,has a generalized T-periodic solution, according to the definition given in the paper by A. Boscaggin, R. Ortega, and L. Zhao [Trans. Amer. Math. Soc. 372 (2019), 677-703]. The proof relies on variational arguments
Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential
Full text linkWe consider the Lorentz force equation in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem
Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics
No full textWe prove the existence of a pair of positive T -periodic solutions
as well as the existence of positive subharmonic solutions of any
order and the presence of chaotic-like dynamics for the scalar
second order ODE
u +aλ,μ(t)g(u) = 0,
where g(x) is a positive function on R
+, superlinear at zero and
sublinear at infinity, and aλ,μ(t) is a T -periodic and sign indefinite
weight of the form λa+
(t)−μa−
(t), with λ,μ > 0 and large
A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth
Full text linkLet 1 < p < +∞ and let Ω C RN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type 0,rm in Ω ,quad partialν u = 0rm on Ω.]]>We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions
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