Boundedness problem for solutions of semilinear asymmetric equations. (Q1412490)

The aim of this paper is to investigate the boundedness of solutions for the semilinear equation \[ (\varphi_p(x'))'+ (p-1)[\alpha\varphi_p(x^+)- \beta\varphi_p(x^-)]+ g(x)= e(t), \tag{1} \] where \(\varphi_p(u)= | u|^{p-2} u\), \(p\geq 2\), \(x^+= \max\{\pm x,0\}\), \(\alpha\), \(\beta\) are positive numbers satisfying \[ \alpha^{-(1/p)}+ \beta^{-(1/p)}= 2\omega^{-1},\quad \omega\in \mathbb{R}^+, \] \(q\in C^\infty(\mathbb{R})\) is semilinear and \(e\in C^\infty(\mathbb{R}/\mathbb{Z})\). The main result states that if \[ aG(x)\leq xg(x),\quad 0\leq x^2 g'(x)\leq (b-1) xg(x), \] for some constants \(1< a\leq b< p\) and \(zb< p+ a\), where \(G(x)= \int^x_0 g(s)\,ds\) and \[ | x^k g^{(k)}(x)|\leq C| g(x)|,\quad k\geq 2, \] then every solution of (1) is bounded, that is, if \(x(t)\) is a solution of (1) then it is defined in \(\mathbb{R}\) and \[ \sup_{t\in\mathbb{R}} (| x(t)|+ | x'(t)|)< +\infty. \] The proof of this result is based on a variant of Moser's twist theorem guaranting the existence of certain invariant curves which lead to a bound of the solutions.