Bounded and Unbounded Motions in Asymmetric Oscillators at Resonance

We consider the boundedness and unboundedness of solutions for the asymmetric oscillator $$\begin{aligned} x''+ax^+-bx^-+g(x)=p(t), \end{aligned}$$ where $$x^+=\max \{x,0\},x^-=\max \{-x,0\}, a$$ and $$b$$ are two positive constants, $$ p(t)$$ is a $$2\pi $$ -periodic smooth function and $$g(x)$$ satisfies $$\lim _{|x|\rightarrow +\infty }x^{-1}g(x)=0$$ . We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. It turns out that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions $$\Phi _p(\theta )$$ and $$\Lambda (h)$$ . Some explicit conditions are given for the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function $$g(x)$$ is required to be a bounded function with asymptotic limits, here we allow $$g(x)$$ be unbounded or oscillatory without asymptotic limits.