Positive solutions to indefinite Neumann problems when the weight has positive average

We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$u" + q(t)g(u) = 0, \quad t \in [0, T],$$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$x' = y, \qquad y' = h(x)y^2 + q(t),$$ with $h(x)$ a continuous function defined on the whole real line.Comment: 17 pages, 3 figure

Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

We study the second-order nonlinear differential equation u\u2032\u2032+a(t)g(u)=0 , where g is a continuously differentiable function of constant sign defined on an open interval I 86R and a(t) is a sign-changing weight function. We look for solutions u(t) of the differential equation such that u(t) 08I, satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for I=R+0 and g(u) 3c 12u 12\u3c3, as well as the case of exponential nonlinearities, for I=R and g(u) 3cexp(u) . The proofs are obtained by passing to an equivalent equation of the form x\u2032\u2032=f(x)(x\u2032)2+a(t)

Uniqueness of positive solutions for boundary value problems associated with indefinite \u3c6-Laplacian-type equations

This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the \u3c6-Laplacian equation 'Equation Presented', where \u3c6 is a homeomorphism with \u3c6(0) = 0, a(t) is a stepwise indefinite weight and g(u) is a continuous function. When dealing with the p-Laplacian differential operator \u3c6(s) = |s|p-2s with p > 1, and the nonlinear term g(u) = u\u3b3 with \u3b3 08 \u211d, we prove the existence of a unique positive solution when \u3b3 \u3f5 ]- 1e, (1 - 2p)/(p - 1)] 2a ]p - 1, + 1e[

Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions

We prove the existence of multiple positive BV-solutions of the Neumann problem \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -\lambda u + u^p$, for $\lambda > 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method.Comment: 28 pages, 1 figur

Unbounded Solutions to Systems of Differential Equations at Resonance

We deal with a weakly coupled system of ODEs of the type xj\u2032\u2032+nj2xj+hj(x1,\u2026,xd)=pj(t),j=1,\u2026,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 \u3c0-periodic, nj 08 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, \u2026 , hd are assumed