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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.

##### MSC:
 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)
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