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′)′−rf(x,u)u=0,x∈(0,l),u′(0)cosα−(pu″)(0)sinα=0,u(0)=0,u′(l)cosγ−(pu″)(l)sinγ=0,u(l)=0, where 0⩽α,γ⩽π/20⩽α,γ⩽π/2, p∈C2[0,l]p∈C2[0,l] and q∈C1[0,l]q∈C1[0,l]. The proof of our main result is based upon bifurcation techniques.