Consider the differential inclusion \[ -\frac{d}{dt}(\| \dot{u}(t)\| ^{p-2}\dot{u}(t))\in\partial F(t,u(t))\quad \text{ a.e. }\, t\in \mathbb{R}, \tag{1} \] where \(p>1, F:\mathbb{R}\times\mathbb{R}^{n}\to \mathbb{R}\) is \(T\)-periodic (\(T>0\)) in \(t\) for all \(x\in\mathbb{R}^{n}\), and \(\partial\) denotes the Clarke subdifferential.
Theorem. Let \(F(t,x)\) be integrable in \(t\) over \([0,T]\) for each \(x\in \mathbb{R}^{n}\) and is locally Lipschitz in \(x\) for each \(t\in [0,T].\) It is supposed that the following conditions hold:
(i) there exist \(c_{1},c_{2} >0\) and \(\alpha\in[0,p-1)\) such that \(\zeta\in\partial F(t,x)\Rightarrow \| \zeta\| \leq c_{1}\| x\| ^{\alpha} +c_{2} \) for all \(x\in\mathbb{R}^{n}\) and a.e. \(t\in[0,T];\)
(ii) there exists \(\gamma\in L^{1}(0,T)\) such that \(F(t,x)\geq\gamma(t)\) for all \(x\in\mathbb{R}^{n}\) and a.e. \(t\in[0,T];\)
(iii) there exists a subset \(E\) of \([0,T]\) with meas \((E)>0\) such that \(\| x\| ^{-q\alpha}F(t,x)\to +\infty\) as \(\| x\| \to \infty\) for a.e. \(t\in E\), where \(q= p/(p-1).\)
Then the differential inclusion (1) has a \( kT\)-periodic solution \(u_{k}\in W^{1,p}_{kT}\) for every positive integer \(k\) such that \(\| u_{k}\| _{\infty}\to\infty\) as \(k\to\infty\).