The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz-Minkowski space { 12div( 07u/ 1a1 12| 07u|\ub2)=f(x,u, 07u) in \u2126, u=0 on 02\u2126 . The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann\u2013Lema\ueetre\u2013Robertson\u2013Walker, as well as Schwarzschild\u2013Reissner\u2013Nordstr\uf6m, spacetimes

The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions

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

Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem

3siWe study the structure of the set of the positive regular solutions of the one-dimensional quasilinear Neumann problem involving the curvature operator $$left({u'}/{sqrt{1+(u')^2}} ight)' = lambda a(x) f(u), quadu'(0)=0,,,u'(1)=0.$$ Here $lambda in R$ is a parameter, $a in L^1(0,1)$ changes sign, and $fin mc{C}(R)$. We focus on the case where the slope of $f$ at $0$, $f'(0)$, is finite and non-zero, and the potential of $f$ is superlinear at infinity, but also the two limiting cases where $f'(0)=0$, or $f'(0)=+infty$, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.partially_openopenLópez-Gómez, Julián; Omari, Pierpaolo; Rivetti, SabrinaLópez Gómez, Julián; Omari, Pierpaolo; Rivetti, Sabrin

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, 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

Qualitative analysis of a curvature equation modelling MEMS with vertical loads

We investigate existence, multiplicity and qualitative properties of the solutions of the Dirichlet problem for a singularly perturbed prescribed mean curvature equation, which appears in the theory of micro-electro-mechanical systems (MEMS) when the effects of capillarity and vertical forces are taken into account

A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, end{equation*} in a bounded Lipschitz domain $Omega subset RR^N$, with $a,b>0$ parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem

Positive solutions of indefinite logistic growth models with flux-saturated diffusion

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator - div (del u/root 1+vertical bar del u vertical bar(2)) - lambda a(x)f(u) in Omega, u - 0 on partial derivative Omega, with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Omega is a bounded domain in R-N with a regular boundary partial derivative Omega, lambda > 0 represents a diffusivity parameter, a is a continuous weight which may change sign in Omega, and f: [0, L] -> R, with L > 0 a given constant, is a continuous function satisfying f(0) = f (L) = 0 and f (s) > 0 for every s is an element of [0, L]. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying lambda. Typically, the solutions we find are regular as long as lambda is small, while as a consequence of the saturation of the flux they may develop singularities when A becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f (s) = s(L - s) and a 1, having no similarity with the case of linear diffusion based on the Fick-Fourier's la