One-dimensional prescribed mean curvature equation with exponential nonlinearity

We establish the existence and multiplicity results for the one-dimensional prescribed mean curvature equation −(u′/1+(u′)2)′=λeu,u(0)=0,u(L)=0. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the length LL of the interval and the value of the parameter λλ.