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Mountain pass type solutions for discontinuous perturbations of the vector \(p\)-Laplacian. (English) Zbl 1102.34008

For a given function \(j:\mathbb R^N\times \mathbb R^N\to (-\infty,+\infty]\) proper (i.e., \(D(j):= \{z\in \mathbb R^N\times \mathbb R^N : j(z)<+\infty\}\not= \emptyset\)), convex and lower semicontinuous, the authors consider the following two-point boundary value problem involving the vector \(p\)-Laplacian operator
\[ -[h_p(u')]' +\varepsilon h_p(u)\in \bar \partial F(t,u)\quad \text{ in\;} [0,T], \tag{1} \]
\[ (h_p(u')(0), -h_p(u')(T))\in \partial j(u(0),u(T)), \tag{2} \]
where \(\varepsilon \geq 0\) is a constant. Here, \(\bar \partial F(t,\eta)\) is the generalized Clarke gradient of the locally Lipschitz function \(F(t,\cdot)\) at \(\eta \in \mathbb R^N\), while \(\partial j\) stands for the subdifferential of \(j\) in the sense of convex analysis.
The aim of the paper is to study the existence of solutions to problem (1),(2). By a solution of the differential inclusion system (1) they understand a function \(u:[0,T]\to \mathbb R^N\) of class \(C^1\) with \(h_p(u')\) absolutely continuous, which satisfies
\[ -[h_p(u')(t)]' +\varepsilon h_p(u(t))\in \bar \partial F(t,u(t)), \quad \text{ for\;a.e.\;} t\in [0,T]. \]
The main result of the paper is an existence theorem of Ambrosetti-Rabinowitz type.

MSC:

34A60 Ordinary differential inclusions
34A36 Discontinuous ordinary differential equations
49J52 Nonsmooth analysis
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