zbMATH — the first resource for mathematics

Multiplicity of solutions of variational systems involving \(\phi\)-Laplacians with singular \(\phi\) and periodic nonlinearities. (English) Zbl 1260.34076
Summary: Using a Lusternik-Schnirelman type multiplicity result for some indefinite functionals due to Szulkin, the existence of at least \(n+1\) geometrically distinct \(T\)-periodic solutions is proved for the relativistic-type Lagrangian system \[ (\phi(q'))'+\nabla_q F(t, q)= h(t), \] where \(\phi\) is an homeomorphism of the open ball \(B_a\subset\mathbb{R}^n\) onto \(\mathbb{R}^n\) such that \(\phi(0)= 0\) and \(\phi=\nabla\Psi\), \(F\) is \(T_j\)-periodic in each variable \(q_j\) and \(h\in L^s(0,T;\mathbb{R}^n)\) \((s>1)\) has mean value zero. Application is given to the coupled pendulum equations \[ \Biggl({q_j'\over \sqrt{1-\| q\|^2}}\Biggr)'+ A_j\sin q_j= h_j(t)\quad (j= 1,\dots, n). \] Similar results are obtained for the radial solutions of the homogeneous Neumann problem on an annulus in \(\mathbb{R}^n\) centered at \(0\) associated to systems of the form \[ \nabla\cdot\Biggl({\nabla w_i\over \sqrt{1-\sum^n_{j=1}\|\nabla w_j\|^2}}\Biggr)+ \partial_{w_j} G(\| x\|, w)= h_i(\| x\|)\;(i= 1,\dots, n), \] involving the extrinsic mean curvature operator in a Minkowski space.

34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
70H05 Hamilton’s equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
Full Text: DOI