Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

Abstract

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem\begin{equation*}-{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = \lambda f(x,u)\mbox{\, in $\Omega$},\qquadu=0 \mbox{\, on $\partial \Omega$},\end{equation*}in a general bounded domain $\Omega\subset\RR^N$, depending on the behaviour at zero or at infinity of $f(x,s)$, or of its potential $F(x,s)=\int_0^s f(x,t)\,dt$. Our main effort here is to describe, in a way as exhaustive as possible, all configurations of the limits of $F(x,s)/s^2$ at zero and of $F(x,s)/s$ at infinity, which yield the existence of one, two, three or infinitely many positive solutions. Either strong, or weak, or bounded variation solutions are considered. Our approach is variational and combines critical point theory, the lower and upper solutions method and elliptic regularization

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space

“…Our results should be compared with similar ones obtained in [8] for a class of semilinear problems, and in [9] and in [14] for a class of quasilinear problems driven by the p-Laplace operator and the mean curvature operator in Euclidean space, respectively. In these papers some kinds of local analogues to the classical conditions of ''sublinearity'' and of ''superlinearity'' have been introduced, extending in various directions some of the results proved in the celebrated work by Ambrosetti, Brezis and Cerami [1].…”

Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space

“…We prove that I has always a minimizer on C[0, R] for any continuous nonlinearity not necessarily positive, and, in our particular case, we provide a sufficient condition ensuring the nontriviality of the minimizers. We note that a variational approach has been also employed in [28] to prove various existence and multiplicity results concerning positive solutions of Dirichlet problems with mean curvature operators in the Euclidean space.…”

Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions

“…2.1] or [21], it is known that (4.5) and (4.6) have at least one solution for all not too large λ > 0. With the help of Lemma 4.1, we further obtain the following results.…”

“…For example, replace a(s) = 1 in the mean curvature operator (see e.g. [11,21]). This approach is suitable to investigate small solutions of the original problem.…”