Affiliation:
1. Department of Mathematics, Pusan National University , Busan 46241 , Republic of Korea
2. School of Mathematics and Statistics, HNP-LAMA, Central South University , Changsha , Hunan 410083 , P. R. China
Abstract
Abstract
In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space:
(P)
u
′
1
+
∣
u
′
∣
2
′
+
h
(
x
)
f
(
u
)
=
0
,
−
1
<
x
<
1
,
u
(
−
1
)
=
u
(
1
)
=
0
,
\left\{\begin{array}{l}{\left(\frac{u^{\prime} }{\sqrt{1+{| u^{\prime} | }^{2}}}\right)}^{^{\prime} }+h\left(x)f\left(u)=0,\hspace{1em}-1\lt x\lt 1,\hspace{1.0em}\\ u\left(-1)=u\left(1)=0,\hspace{1.0em}\end{array}\right.
where
h
∈
C
1
(
[
−
1
,
1
]
)
h\in {C}^{1}\left(\left[-1,1])
and
f
∈
C
1
(
[
0
,
∞
)
;
[
0
,
∞
)
)
f\in {C}^{1}\left(\left[0,\infty );\hspace{0.33em}\left[0,\infty ))
. Under suitable conditions on
h
h
and monotone condition on
f
(
s
)
s
\frac{f\left(s)}{s}
, by introducing a modified Picone-type identity, we prove that the problem has at most one positive symmetric solution.
Subject
General Mathematics,Statistical and Nonlinear Physics