Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II

Abstract

We consider non-classical solutions of the quasilinear boundary value problem { − ( u ′ 1 + ( u ′ ) 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = u ( L ) = 0 , where λ and L are positive parameters. We give complete descriptions of the structure of bifurcation curves and determine the exact numbers of positive non-classical solutions of the model problems for various nonlinearities f ( u ) = e u , f ( u ) = ( 1 + u ) p ( p > 0 ) , f ( u ) = e u − 1 , f ( u ) = u p ( p > 0 ) , and f ( u ) = a u ( a > 0 ) . The methods used are elementary and based on a detailed analysis of time maps. Moreover, for the case f ( u ) = | u | p − 1 u ( p > 0 ) , we also obtain the exact number of all sign-changing non-classical solutions and show the global structure of bifurcation curves.