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
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
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
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.
We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problemdepending on the behaviour at the origin and at infinity of the function f. We consider solutions that are possibly discontinuous at the points where they attain the value zero.
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