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Hartman-type results for \(p(t)\)-Laplacian systems. (English) Zbl 1025.34017
Summary: Consider the weighted \(p(t)\)-Laplacian ordinary system \[ -\biggl(w(t) \bigl|u'(t)\bigr |^{p(t)-2}u'(t)\biggr)'+ w(t)f\bigl(t, u(t) \bigr)=0\text{ in }(a,b),\;u(a)= u(b)=0, \] with \(f\in C([a,b]\times \mathbb{R}^N, \mathbb{R}^N)\), \(w\in C([a,b], \mathbb{R})\), \(p\in C([a,b],\mathbb{R})\) and \(p(t)>1\) for \(t\in [a,b]\). It is proved that if \(\exists R>0\) such that \(\langle f(t,u),u\rangle\geq 0\), \(\forall t\in[a,b]\), \(\forall u\in\mathbb{R}^N\) with \(|u|=R\), then the problem has a solution \(u\) such that \(|u(t)|\leq R\) for \(t\in[a,b]\). As a corollary of this result, taken \(w(t)=t^{n-1}\), the authors obtain the existence of the radial solutions for elliptic systems. This result generalizes the corresponding results obtained by Hartman and Mawhin.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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