The relationship between two-parameter perturbed sine-Gorden type equation and nonlinear Klein-Gordon equation

Abstract

We consider the two-parameter nonlinear eigenvalue problem of perturbed sine-Gordon type: u(t) + μu(t) = λg(u(t)), u(t)>0 in I:= (0,1), u(0) = u(1) = 0, where p, A > 0 are parameters and g(u) = a 1 u - a 2 u p + o(u p ) as u 1 0 (p > 3, a 1 , a 2 > 0). This equation is called a perturbed sine-Gordon type equation when g(u) = u + sin u. By using a variational method on general level sets, we establish the different types of asymptotic formulas for the solutions as p - co for the case p > 5, p = 5, 3 < p < 5, and p = 3, respectively. We emphasize that critical exponents are p = 3, 5 and only in the case where p = 3, the solution of the above equation is related asymptotically to that of the associated nonlinear stationary Klein-Gordon equation as μ → ∞.