A variational method for multivalued boundary value problems. (English) Zbl 1465.34032

The paper is concerned with a potential system \[-\left [\phi (u') \right ]' +\varepsilon \phi (u) = \nabla_u F(t,u) \quad \mbox{ in } (0,T), \] subject to a boundary condition of type \[\left ( \phi(u'(0)), -\phi(u'(T))\right ) \in \partial j (u(0), u(T)),\] where \(\phi: \mathbb{R}^N \to \mathbb{R}^N\) is a monotone homeomorphism with a particular structure – including the vector \(p\)-Laplacian \(\phi_p\), and \(j:\mathbb{R}^N \times \mathbb{R}^N \to (-\infty, +\infty]\) is a convex, lower semicontinuous function. Using the approach introduced by G. Moroşanu and the reviewer [J. Math. Anal. Appl. 313, No. 2, 738–753 (2006; Zbl 1105.34008)] for the \(\phi_p\) case, the authors obtain the existence of solutions when the corresponding energy functional is coercive.


34B15 Nonlinear boundary value problems for ordinary differential equations
47J30 Variational methods involving nonlinear operators
37C60 Nonautonomous smooth dynamical systems


Zbl 1105.34008
Full Text: DOI


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