Asymptotic analysis of the first eigenvalues for sturm-liouville problems with applications

The present paper is concerned with the Sturm-Liouville eigenvalue problems. The accurate asymptotic behavior of the first eigenvalues on the jump set is given. The results indicate that the asymptotic behaviors are only related to the values of the coefficients of the equations in the neighborhood of the end points. With an application on the nonlinear propagation phenomena, we discuss the monotonicity of the first eigenvalues with respect to the parameter involved in both equations and boundary conditions. The infimum of the parameter which guarantees the monotonicity of the first eigenvalues is obtained.