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EnglishWe study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation $$ \begin{cases} \displaystyle -\text{div}\bigg(\frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\bigg)=f(|x|,v) & \text{in } B_R,\\v=0 & \text{on } \partial B_R, \end{cases} $$ where $B_R$ is a ball in $\mathbb{R}^N$ ($N\ge 2$). 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.
| Titolo: | POSITIVE RADIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE MINKOWSKI-CURVATURE EQUATION IN A BALL |
| Autori: | |
| Data di pubblicazione: | 2014 |
| Stato di pubblicazione: | Pubblicato |
| Rivista: | |
| Abstract: | We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation $$ \begin{cases} \displaystyle -\text{div}\bigg(\frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\bigg)=f(|x|,v) & \text{in } B_R,\\v=0 & \text{on } \partial B_R, \end{cases} $$ where $B_R$ is a ball in $\mathbb{R}^N$ ($N\ge 2$). 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. |
| Handle: | http://hdl.handle.net/11368/2921638 |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.12775/TMNA.2014.034 |
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