Subharmonic solutions of the prescribed curvature equation

Abstract

We study the existence of subharmonic solutions of the prescribed curvature equationAccording 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.

Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball

“…The conclusion then follows from the Poincaré-Birkhoff fixed point theorem. We also mention that, combining the arguments in the present paper with the ones in [13,Section 3] it should be possible to prove the existence of pairs of subharmonic solutions (namely, mT -periodic solutions for m > 1) for the equation u 1 − (u ) 2 + q(t)g(u) = 0.…”

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

“…The theorems in [32] apply to higher dimensional Hamiltonian systems as well. For another recent application of such resuts to planar systems, in which the uniqueness of the solutions of the Cauchy problems is not required, see also [18]. In our case, even if we apply the Fonda-Ureña theorem, we still need to assume at least an upper bound on g(x)/x and h(y)/y near zero, so to avoid the possibility that a (nontrivial) solution u(•) of (2.1) with u(0) = (0, 0) may hit the origin at some time t ∈ ]0, T ], thus preventing the rotation number to be well defined.…”

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

“…The crucial point, here, is to define, for a (possibly discontinuous) BV-solution, a suitable notion of intersection with the constant u 0 , so as to provide their multiplicity. To the best of our knowledge, a similar issue has never been faced in the context of mean curvature equations, for which only few high multiplicity results are known [44,26,45,46,48,18].…”