A critically degenerate elliptic Dirichlet problem, spectral theory and bifurcation

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)