In this paper we will use the degree theory to give an existence result to the boundary value problem of nonlinear perturbations of the scalar p-Laplacian with the multi-point boundary condition involving the Lebesgue-Stieltjes integral. Among the boundary conditions considered in this paper, we will find also the optimal bounds on the growth of the nonlinear perturbations in our existence result.

Keywords: p-Laplacian; Multi-point boundary condition; Lebesgue-Stieltjes integral; Degree theory; Eigenvalue; Lp norms