### Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball

#### Abstract

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation

$$

\cases

\displaystyle

-\text{\rm div}\bigg( \frac{\nabla v} {\sqrt{1 - |\nabla v|^2}}\bigg)= f(|x|,v) &\quad \text{in } B_R,

\\

\displaystyle

v=0 & \quad \text{on } \partial B_R,

\endcases

$$

< p> 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.

$$

\cases

\displaystyle

-\text{\rm div}\bigg( \frac{\nabla v} {\sqrt{1 - |\nabla v|^2}}\bigg)= f(|x|,v) &\quad \text{in } B_R,

\\

\displaystyle

v=0 & \quad \text{on } \partial B_R,

\endcases

$$

< p> 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.

#### Keywords

Quasilinear elliptic differential equation; Minkowski-curvature; Dirichlet boundary condition; radial solution; positive solution; existence; multiplicity; variational methods

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