Boundedness of solutions for equations with \(p\)-Laplacian and an asymmetric nonlinear term (Q703827)

Sufficient conditions are obtained for the boundedness of solutions and existence of quasi-periodic solutions of differential equations of the form \[ (\phi_p(x'))'+\alpha\phi_p(x^+)-\beta\phi_p(x^-)=f(t,x),\tag{\(*\)} \] where \(x^+=\max\{x,0\}\), \(x^-=\max\{-x,0\}\), \(\phi_p(s)=|s|^{p-2}s,p>1\) and \(\alpha,\beta\) are strictly positive constants, \(f\) is smooth and \(2\pi\) periodic in \(t\). The author has reviewed the development of the work in this direction leading to the present study. The main result of the paper runs as follows: Suppose that \(f\in C^{7,6}(S^1\times \mathbb{R})\), \(\lim_{x\to\pm\infty}f(t,x)=f_\pm(t)\), uniformly in \(t\) and \(\lim_{x\to\pm\infty}x^m\frac{\partial^{n+m}}{\partial t^n\partial x^m}f(t,x)=f_{\pm,m,n}(t)\), for \((n,m)=(0,6)\), (7,0) and (7,6). Let \(f_{\pm,m,n}(t)\equiv 0\) for \(m=6, n=0,7\). If \[ Z(\theta)=\int_{\{t\in [0,\frac{2\pi}{n}]: \nu(t)>0\}} f_+(t+\theta)\nu(t)dt +\int_{\{t\in [0,\frac{2\pi}{n}]: \nu(t)<0\}} f_-(t+\theta)\nu(t)dt \] is of constant sign, then all solutions of (\(*\)) are defined in \(\mathbb{R}\) and for each \(x(t)\), we have \(\sup_{t\in \mathbb{R}}\{|x(t)|+|x'(t)|\}<+\infty\). Moreover, (\(*\)) admits infinitely many subharmonic solutions and quasi-periodic solutions. It is a good paper.