Abstract
In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type
(P)
−
d
i
v
∇
u
1
−
|
∇
u
|
2
=
λ
f
(
u
)
in
B
1
,
u
=
0
on
∂
B
1
,
$$\left\{\begin{aligned}\hfill & -\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u}{\sqrt{1-\vert \nabla u{\vert }^{2}}}\right)=\lambda f(u)\quad \text{in}\enspace {B}_{1},\hfill \\ \hfill & u=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \text{on}\enspace \partial {B}_{1},\enspace \hfill \end{aligned}\right.$$
where λ is a positive parameter,
B
1
=
{
x
∈
R
N
:
|
x
|
<
1
}
${B}_{1}=\left\{x\in {\mathbb{R}}^{N}:\vert x\vert {< }1\right\}$
, f : [0, ∞) → [0, ∞) is a continuous function. Using the fixed point index in a cone, we prove the results on both uniqueness and multiplicity of positive radial solutions of (P).
Entire radial solutions of elliptic systems and inequalities of the mean curvature type