American Institute of Mathematical Sciences

November  2012, 32(11): 4015-4026. doi: 10.3934/dcds.2012.32.4015

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

 1 Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium

Received  April 2011 Revised  June 2011 Published  June 2012

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.
Citation: Jean Mawhin. Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4015-4026. doi: 10.3934/dcds.2012.32.4015
References:
 [1] C. Bereanu, P. Jebelean and J. Mawhin, Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular $\phi$-Laplacians, Matemática Contemporânea, 36 (2009), 51-65. [2] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynamics Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3. [3] C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbations of singular $\phi$-Laplacian, Rendiconti Lincei: Matematica e Applicazioni, 22 (2011), 89-111. doi: 10.4171/RLM/589. [4] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities, Discrete Continuous Dynamical Systems, 28 (2010), 637-648. [5] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calculus Variations Partial Differential Equations, to appear. [6] C. Bereanu and P. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., to appear. [7] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. [8] H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Communic. Applied Anal., 15 (2011), 235-250. [9] K.-C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Anal., 13 (1989), 527-537. doi: 10.1016/0362-546X(89)90062-X. [10] G. Fournier, D. Lupo, M. Ramos and M. Willem, Limit relative category and critical point theory, Dynamics Reported, 3 (1994), 1-24. [11] G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259-281. [12] G. Fournier and M. Willem, Relative category and the calculus of variations, in "Variational Methods" (Paris, 1988) (eds. H. Berestycki, J. M. Coron and I. Ekeland), Progr. Nonlinear Differential Equations Appl., 4, Birkhäuser Boston, Boston, MA, (1990), 95-104. [13] J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385. [14] J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. Henri-Poincaré Anal. Non Linéaire, 6 (1989), 415-434. [15] J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287. doi: 10.1016/0022-0396(84)90180-3. [16] J. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, in "Nonlinear Anal. and Optimization" (Bologna, 1982) (ed. C. Vinti), Lecture Notes in Math., 1107, Springer, Berlin (1984), 181-192. [17] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. [18] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132. [19] P. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conf. Ser. in Math., 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 1986. [20] P. Rabinowitz, On a class of functionals invariant under a $Z_n$ action, Trans. Amer. Math. Soc., 310 (1988), 303-311. doi: 10.1090/S0002-9947-1988-0965755-5. [21] M. Reeken, Stability of critical points under small perturbations. I: Topological theory, Manuscripta Math., 7 (1972), 387-411. [22] J. T. Schwartz, Nonlinear Functional Analysis, Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. [23] A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15 (1990), 725-739. doi: 10.1016/0362-546X(90)90089-Y.

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References:
 [1] C. Bereanu, P. Jebelean and J. Mawhin, Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular $\phi$-Laplacians, Matemática Contemporânea, 36 (2009), 51-65. [2] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynamics Differential Equations, 22 (2010), 463-471. doi: 10.1007/s10884-010-9172-3. [3] C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbations of singular $\phi$-Laplacian, Rendiconti Lincei: Matematica e Applicazioni, 22 (2011), 89-111. doi: 10.4171/RLM/589. [4] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities, Discrete Continuous Dynamical Systems, 28 (2010), 637-648. [5] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calculus Variations Partial Differential Equations, to appear. [6] C. Bereanu and P. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., to appear. [7] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810. [8] H. Brezis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Communic. Applied Anal., 15 (2011), 235-250. [9] K.-C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Anal., 13 (1989), 527-537. doi: 10.1016/0362-546X(89)90062-X. [10] G. Fournier, D. Lupo, M. Ramos and M. Willem, Limit relative category and critical point theory, Dynamics Reported, 3 (1994), 1-24. [11] G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 259-281. [12] G. Fournier and M. Willem, Relative category and the calculus of variations, in "Variational Methods" (Paris, 1988) (eds. H. Berestycki, J. M. Coron and I. Ekeland), Progr. Nonlinear Differential Equations Appl., 4, Birkhäuser Boston, Boston, MA, (1990), 95-104. [13] J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372-385. [14] J. Mawhin, Forced second order conservative systems with periodic nonlinearity, Ann. Inst. Henri-Poincaré Anal. Non Linéaire, 6 (1989), 415-434. [15] J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287. doi: 10.1016/0022-0396(84)90180-3. [16] J. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, in "Nonlinear Anal. and Optimization" (Bologna, 1982) (ed. C. Vinti), Lecture Notes in Math., 1107, Springer, Berlin (1984), 181-192. [17] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989. [18] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132. [19] P. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conf. Ser. in Math., 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 1986. [20] P. Rabinowitz, On a class of functionals invariant under a $Z_n$ action, Trans. Amer. Math. Soc., 310 (1988), 303-311. doi: 10.1090/S0002-9947-1988-0965755-5. [21] M. Reeken, Stability of critical points under small perturbations. I: Topological theory, Manuscripta Math., 7 (1972), 387-411. [22] J. T. Schwartz, Nonlinear Functional Analysis, Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. [23] A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., 15 (1990), 725-739. doi: 10.1016/0362-546X(90)90089-Y.
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