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

Authors: Tetsutaro Shibata

Issue: Volume 55 (2006), Issue 5, 1557-1572

Abstract: We consider the two-parameter nonlinear eigenvalue problem of perturbed sine-Gordon type: \begin{multline*} u''(t) + \mu u(t) = \lambda g(u(t)),\\ u(t) > 0 \mbox{ in } I := (0,1), \quad u(0) = u(1) = 0, \end{multline*} where $\mu$, $\lambda > 0$ are parameters and $g(u) = a_{1}u - a_{2}u^{p} + o(u^{p})$ as $u \downarrow 0$ ($p \ge 3$, $a_{1}$, $a_{2} > 0$). 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 $\mu \to \infty$ 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 $\mu \to \infty$.