In an open, bounded subset Omega of R-N such that 0 is an element of Omega we consider the nonlinear eigenvalue problem -Sigma(N)(i,j,=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + V(x)u + n(x,del u)+ g(x, u) = lambda u in Omega integral(Omega) u(2) + Sigma(N)(i,j=1) A(ij)partial derivative(j)u partial derivative(i)u dx < infinity and u = 0 on partial derivative Omega, where V is an element of L-infinity(Omega) and the nonlinear terms n and g are of higher order near 0 so that the formal linearization about the trivial solution u 0 is - Sigma(N)(i,j=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + Vu = lambda u. The leading term is degenerate elliptic on Omega because it is assumed that there are constants C-2 >= C-1 > 0 such that C1 vertical bar x vertical bar(2)vertical bar xi vertical bar(2) <= Sigma(N)(i,j=1) A(ij)(x)xi(j)xi(i) <= C-2 vertical bar x vertical bar(2)vertical bar xi vertical bar(2 )for all xi is an element of R-N and almost all x is an element of Omega. This is the lowest level of degeneracy at x = 0 for which the linearization has a non-empty essential spectrum. Furthermore, elliptic regularity theory does not apply at x = 0. Eigenfunctions of the linearization and solutions of the nonlinear problem having finite energy may be singular at the origin. The main results establish conditions for the existence or not of eigenvalues of the linearization, describe the behaviour of eigenfunctions as x -> 0 and determine values of the parameter lambda at which bifurcation from the line of trivial solutions occurs. Standard bifurcation theory does not apply, even when n and g are smooth functions, since the nonlinear terms generate operators which are Gateaux but not Frechet differentiable at the trivial solution. (C) 2019 Elsevier Ltd. All rights reserved.