Asymptotic formula for eigenvalues of simple pendulum problems

We consider the nonlinear eigenvalue problem -Deltau=lambdaf(u), u>0 in B-R, u=0 on partial derivativeB(R), where B-R is a ball with radius R>0 and lambda>0 is a parameter. Under the appropriate conditions of f, it is known that for a given 0<epsilon<1, there exists (lambda,u)=(lambda(epsilon),u(epsilon)) satisfying the equation with integral(BR)F(u(epsilon)(x)) dx=\B-R\F(u(0))(1-epsilon), where F(u)=integral(0)(u)f(s) ds and u(0)>0 is the smallest zero of f in R+. Furthermore, u(epsilon)(x)-->u(0) (xis an element ofB(R)) and lambda(epsilon)-->infinity as epsilon-->0. This concept of parametrization of solution pair by a new parameter epsilon is based on the variational structure of the equation. We establish the asymptotic formulas for lambda(epsilon) as epsilon-->0 with the 'optimal' estimate of the second term.