Classical and non-classical solutions of a prescribed curvature equation

Abstract

We discuss existence and multiplicity of positive solutions of the \ud one-dimensional prescribed curvature problem \ud $$\ud -\left(\ud {u'}/{\sqrt{1+{u'}^2}}\right)' = \lambda f(t,u),\ud \quad\ud u(0)=0,\,\,u(1)=0, \ud $$\ud depending on the behaviour at the origin and at infinity of the potential $\int_0^u f(t,s)\,ds$. Besides solutions in $W^{2,1}(0,1)$, we also consider solutions in $W_{loc}^{2,1}(0,1)$ which are possibly discontinuos at the endpoints of $[0,1]$. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem

Subharmonic solutions of the prescribed curvature equation

“…for any s ∈ R. By definition ofψ and by conditions (16) and (18), the only equilibrium point of (19) and (20) is (0, 0) and all level curves of E G and E H are closed curves around (0, 0). Hence global existence and uniqueness of solution hold for every Cauchy problem associated with (19) and (20).…”

“…Hence global existence and uniqueness of solution hold for every Cauchy problem associated with (19) and (20). Let us introduce two auxiliary functions M ∓ : R 2 \ {0} → R, defined by …”

“…The coexistence of classical and non-classical solutions of (1), according to the terminology introduced in [18,19,15,17,20], is determined by the specific structure of the curvature operator u / 1 + u 2 , which behaves like the 2-Laplacian u near zero and like the 1-Laplacian sgn(u ) at infinity. These considerations lead us to introduce the following concept of periodic solution for equation (1) that will be considered throughout this paper.…”

Steady state solutions of a bi-stable quasi-linear equation with saturating flux

“…Here the bifurcation is again a subcritical pitchfork, but the classical solutions stop existing before we reach a saddle-node; see Figure 5. 3 6 Non-classical Solutions to the Problem Non-classical solutions for problems related to the prescribed mean curvature equation have been discussed, to some extent in [6]. However, in that paper the non-classical solutions are C ∞ ((0, L)).…”

“…This boundary value problem, for different choices of the nonlinearity f (u) and boundary conditions has received attention from a variety of authors including Pan [5], Bonheure et al [6] and Habets and Omari [7]. In [7], Habets and Omari study (1.3) with Dirichlet boundary conditions, taking f (u) = (u p ) + , for p > 0, and they investigate the influence of the concavity of this choice of f (u) on the multiplicity of solutions to the problem.…”