Nodal solutions of a nonlinear eigenvalue problem of the Euler–Bernoulli equation

We are concerned with determining values of r, for which there exist nodal solutions of the boundary value problems of the Euler–Bernoulli equation ( p ( x ) u ″ ) ″ − ( q ( x ) u ′ ) ′ − r f ( x , u ) u = 0 , x ∈ ( 0 , l ) , u ′ ( 0 ) cos α − ( p u ″ ) ( 0 ) sin α = 0 , u ( 0 ) = 0 , u ′ ( l ) cos γ − ( p u ″ ) ( l ) sin γ = 0 , u ( l ) = 0 , where 0 ⩽ α , γ ⩽ π / 2 , p ∈ C 2 [ 0 , l ] and q ∈ C 1 [ 0 , l ] . The proof of our main result is based upon bifurcation techniques.