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

Isabel Coelho, Chiara Corsato, Sabrina Rivetti

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.

Keywords


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

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