A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem

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.