On a sub-supersolution method for the prescribed mean curvature problem

Le, Vy Khoi

doi:10.1007/s10587-008-0034-7

Cited by 16 publications

(18 citation statements)

References 36 publications

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“…Proceeding as above we find a solution u 2 ∈ H 1 0 (Ω) ∩ L ∞ (Ω) of (27) such that 0 u 2 (x) β 2 (x) in Ω and u 2 = 0. Iterating this argument we obtain a sequence of non-trivial non-negative solutions of (27) such that lim n→+∞ u n L ∞ (Ω) = 0.…”

Section: Potential Oscillatory At Zeromentioning

confidence: 99%

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

Obersnel

Omari

2010

Journal of Differential Equations

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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

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Section: Potential Oscillatory At Zeromentioning

confidence: 99%

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

Obersnel

Omari

2010

Journal of Differential Equations

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“…have been a topic of recent consideration by several authors (see [3,12,17,18,19] and the references there-in). Our main results are now stated.…”

Section: B)mentioning

confidence: 99%

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

Brubaker

Lindsay²

2013

Eur. J. Appl. Math

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The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

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“…We first need some abstract existence theorems for the inequality (1.7) in the coercive case. Although there are existence theorems given in [13] or [22,24] (and the references therein) for related problems, we could not find an exact reference for our problem here. Therefore, a statement of such result together with its proof is given here.…”

Section: Existence Of Solutions In Coercive Problemsmentioning

confidence: 99%

On a sub-supersolution method for the prescribed mean curvature problem

Cited by 16 publications

References 36 publications

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

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

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

Some existence results and properties of solutions in quasilinear variational inequalities with general growths

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