AbstractIn this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation
$$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$
{
−
(
u
′
1
−
u
′
2
)
′
=
λ
f
(
u
)
,
x
∈
(
−
L
,
L
)
,
u
(
−
L
)
=
0
=
u
(
L
)
,
where λ and L are positive parameters, $f\in C[0,\infty ) \cap C^{2}(0,\infty )$
f
∈
C
[
0
,
∞
)
∩
C
2
(
0
,
∞
)
, and $f(u)>0$
f
(
u
)
>
0
for $0< u< L$
0
<
u
<
L
. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies $f''(u)>0$
f
″
(
u
)
>
0
and $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$
u
f
′
(
u
)
≥
f
(
u
)
+
1
2
u
2
f
″
(
u
)
for $0< u< L$
0
<
u
<
L
. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.
Exact multiplicity of positive solutions of a semipositone problem with concave–convex nonlinearity