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

Béhi, Droh Arsène; Adjé, Assohoun

Abstr. Appl. Anal. 2020, Article ID 8463263, 8 p. (2020).

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.

Reviewer: Petru Jebelean (Timişoara)

MSC:

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

Keywords:

nonlinear boundary value problem; lower semicontinuous convex function; critical point

Citations:

Zbl 1105.34008

References:

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