Variational methods for nonlinear multiparameter elliptic eigenvalue problems

We consider the following nonlinear multiparameter problem u ''(r) + N - 1/r u'(r) (n) Sigma(k=1)mu(k)u(r)(Pk) = lambda u(r) 0 < r < 1 u(r) > 0 0 less than or equal to r < 1 u'(0) = 0 u(1) = 0 where N greater than or equal to 3, mu = (mu(1), mu(2),...,mu(n)) is an element of R-+(n) lambda is an element of R+ are parameters and 1 < p(1) < p(2) < ... < p(n) < 1 + 4/N. We shall establish an asymptotic formula of variational eigenvalues lambda = lambda(mu, alpha) as mu(i) --> infinity by using the Ljusternik-Schnirelman theory on the L-2-sphere M-alpha, where alpha > 0 is the radius of the L-2-sphere.