We prove the existence of multiple positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space \ud \begin{equation*}\ud \begin{cases}\ud -{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u, \nabla u) \ud & \hbox{ in } \Omega, \ud \\\ud u=0& \hbox{ on } \partial \Omega. \ud \end{cases}\ud \end{equation*}\ud Here $\Omega$ is a bounded regular domain in $\RR^N$\ud and the function $f=f(x,s,\xi)$ is\ud either sublinear, or superlinear, or sub-superlinear near $s=0$. \ud The proof combines topological and variational methods

Abstract:We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact, they are linear affine transformations of P0 as long as the transformation returns a value in [0, 1]. We study the properties of NL measures that are (at least) capacities, and show that they can be partitioned into three major subfamilies. We investigate their consistency, which ranges from 2-coherence, the minimal condition satisfied by all, to coherence, and the kind of beliefs they can represent. There is a variety of different situations that NL models can incorporate, from generalisations of the Pari-Mutuel model, the ε-contamination model and other models to conflicting attitudes of an agent towards low/high P0-probability events (both prudential and imprudent at the same time), or to symmetry judgments. The consistency properties vary with the beliefs represented, but not strictly: some conflicting and partly irrational moods may be compatible with coherence. In a final part, we compare NL models with their closest, but only partly overlapping, models, neo-additive capacities and probability intervals.

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation 8 > < > :where B R is a ball in R N (N ≥ 2). According to the behaviour of f = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

We discuss existence, uniqueness, regularity and boundary behaviour of solutions of the Dirichlet problem for the prescribed anisotropic mean curvature equation\ud \begin{equation*}\ud {\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}}, \ud \end{equation*}\ud where $a,b>0$ are given parameters and $\Omega$ is a bounded Lipschitz domain in $\RR^N$. \ud This equation appears in the modeling theory of capillarity phenomena for compressible fluids and in the description of the geometry of the human cornea