A variational approach for one-dimensional prescribed mean curvature problems (Q2925686)

The authors discuss the multiplicity of nonnegative solutions of the parametric one-dimensional mean curvature problem NEWLINE\[NEWLINE\begin{aligned} &- \left( \frac{u'}{\sqrt{1+u'{}^2}} \right)' = \lambda f(t,u)\quad \text{in }(0,1),\\ & u(0)=u(1)=0, \end{aligned}NEWLINE\]NEWLINE where \(f : [0, 1] \times \mathbb{R} \to \mathbb{R}\) is an \(L^1\)-Carathéodory function and \(\lambda > 0\) is a real parameter. Their main effort here is to describe the configuration of the limits of a certain function, depending on the potential at zero, that yield, for certain values of the parameter, the existence of infinitely many weak nonnegative and nontrivial solutions. Moreover, thanks to a classical regularity result due to Lieberman, this sequence of solutions strongly converges to zero in \(C^1 ([0, 1])\). Their approach is based on recent variational methods [\textit{B. Ricceri}, J. Comput. Appl. Math. 113, No. 1--2, 401--410 (2000; Zbl 0946.49001)].