A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem (Q6084999)

The authors study the existence of nonnegative symmetric solutions of the following boundary value problem (IVP) \[ -\Bigl(\frac{u'(x)}{\sqrt{1+[u'(x)]^{2}}}\Bigr)'(x)=h(x)f(u(x))\quad\text{ for each }x\in (-1,1) \] subject to the Dirichlet boundary condition: \(u(-1)=u(1)=0\), where \(h\in C^{1}_{+}[-1,1]\) and \(f\in C^{1}_{+}(\mathbb R_{+})\). Under suitable assumptions on \(h\) and monotonicity conditions on \(f(u)/u\). The authors introduce a modified Picone-type identity and prove that the above BVP has at least one nonnegative symmetric solution in \(C[-1,1]\).