variational methods Asymptotic behaviors 2 Asymptotic profiles 2 Eigenvalues and eigenfunctions 2 Elliptic problem 2 Existence of Solutions 2 Fractional Laplace operator 2 Nonlinear eigenvalue problem 2 Positive solution 2 Principal eigenvalues 2 Quasilinear problems 2 Variational methods 2 linear and nonlinear eigenvalue problems 2 perturbative method 2 sub- and super-solutions 2