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Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities

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  • 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 $$(\phi(q'))' + \nabla_qF(t,q) = h(t),$$ 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 $\phi = \nabla \Phi$, $F$ 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 $$\left(\frac{q'_j}{\sqrt{1 - \|q\|^2}}\right)' + A_j \sin q_j = h_j(t) \quad (j = 1,\ldots,n).$$ 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 $$\nabla \cdot \left(\frac{\nabla w_i}{\sqrt{1 - \sum_{j=1}^n \|\nabla w_j\|^2}}\right) + \partial_{w_j} G(\|x\|,w) = h_i(\|x\|), \quad (i = 1,\ldots,n),$$ involving the extrinsic mean curvature operator in a Minkovski space.
    Mathematics Subject Classification: Primary: 34C25, 35J65; Secondary: 58E06, 70H05.

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