Unbounded Solutions to Systems of Differential Equations at Resonance

Journal of Dynamics and Differential Equations - We deal with a weakly coupled system of ODEs of the type $$\begin{aligned} x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots ,x_d) = p_j(t), \qquad...     

Multiple solutions of weakly-coupled systems with p-laplacian operators

We prove a multiplicity result for weakly-coupled systems with p-laplacian operators of the form $$(\phi_{pi}(u_{i}^{\prime}))^{\prime}+gi(u_i)=h_i(t,u,u^\prime),\qquad t\ \in \ (0,1),\qquad$$ Neumann boundary conditions are assumed and the nonlinearity is supposed to be superlinear asymmetric. We use a topological degree method based on a continuation theorem and on the performance of a time-map technique for the unperturbed case.     

A note on the existence of multiple solutions for a class of systems of second order ODEs

We prove the existence of multiple solutions for some systems of second order ODEs with Dirichlet boundary conditions. Such systems are obtained by coupling scalar ODEs with different growth conditions. The proof relies on a global continuation technique.     

On the boundedness of solutions to a nonlinear singular oscillator

We study a second order scalar equation of the form x′′ + V′(x) = p(t), where p is a π-perodic function and V is a singular potential. We give sufficient conditions on V, p ensuring that all solutions are bounded; we prove the existence of Aubry–Mather sets as well.     

Infinitely many radial solutions to a boundary value problem in a ball

In this paper we are concerned with the existence and multiplicity of radial solutions to the BVP whereB is an open ball in ?^K and u??·(a(|?u|)?u) is a nonlinear differential operator (e.g. the plaplacian or the mean curvature operator). The function f is defined in a neighborhood of u=0 and satisfies a «sublinear»-type growth condition for u→0. We use a degree approach combined with a time-map technique. Multiplicity results are obtained also for nonlinearities of concave-convex type.