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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.

34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
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