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

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∈C1([−1,1])h\in {C}^{1}\left(\left[-1,1]) and f∈C1([0,∞);[0,∞))f\in {C}^{1}\left(\left[0,\infty );\hspace{0.33em}\left[0,\infty )). Under suitable conditions on hh 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.

Keywords

• uniqueness
• mean curvature
• symmetry
• positive solutions
• 34a12
• 34b05
• 35j66
• 35j93