Periodic solutions of a wave equation with concave and convex nonlinearities
The paper deals with the following problem describing the periodic solutions of a nonautonomous wave equation \[ u_{tt}-u_{xx}=f_{\lambda\mu}(x,t,u), \quad\text{for }0<x<\pi,\;t\in \mathbb{R}, \] \[ u(0,t)=u(\pi,t)=0,\quad \text{for }t\in \mathbb{R},\qquad u(x,t+2\pi)=u(x,t),\quad \text{for }t\in \mathbb{R}. \] The nonlinearity \(f_{\lambda\mu}\in C([0,\pi]\times \mathbb{R}^2,\mathbb{R})\) has the form \(f_{\lambda\mu}=\lambda g(x,t,\xi)+\mu h(x,t,\xi)\) with \(\lambda,\mu\in \mathbb{R}\), \(g\) superlinear in \(\xi\), \(h\) sublinear in \(\xi\) and both \(g\), \(h\) \(2\pi\)-periodic in \(t\) and nondecreasing in \(\xi\). If \(\lambda\) and \(\mu\) have the same sign then \(f_{\lambda\mu}\) is monotone. The paper treats also the case \(\lambda\mu<0\) where this monotonicity is lost. Using variational methods the existence and multiplicity of periodic solutions of this problem are studied for the model nonlinearity \(f_{\lambda\mu}=a(x,t) (\lambda | \xi| ^{p-2}\xi+\mu| \xi| ^{q-2}\xi) \) with \(p>2>q>1\) and \(a\in C([0,\pi]\times \mathbb{R},\mathbb{R})\) positive and \(2\pi\)-periodic in \(t\) and in the general case.
existence and multiplicity of periodic solutions
scientific article; zbMATH DE number 1278769
scientific article