Abstract
In this paper we survey some recent results on the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of the mean extrinsic curvature operator in Minkowski space. Our approach relies on the Leray-Schauder degree, upper and lower solutions, and critical point theory.
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References
R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982-83), 131–152.
Bereanu C., Jebelean P., Mawhin J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Amer. Math. Soc. 137, 171–178 (2009)
Bereanu C., Jebelean P., Mawhin J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283, 379–391 (2010)
Bereanu C., Jebelean P., Mawhin J.: Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular \({\phi}\)-Laplacians. Matemática Contemporânea 36, 51–65 (2009)
Bereanu C., Jebelean P., Mawhin J.: Periodic solutions of pendulum-like perturbations of singular and bounded \({\phi}\)-Laplacians. J. Dynamics Differential Equations 22, 463–471 (2010)
Bereanu C., Jebelean P., Mawhin J.: Radial solutions for Neumann problems with \({\phi}\)-Laplacians and pendulum-like nonlinearities. Discrete Contin. Dynam. Systems-A 28, 637–648 (2010)
C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, submitted.
Bereanu C., Jebelean P., Mawhin J.: Variational methods for nonlinear perturbations of singular \({\phi}\)-Laplacians. Rend. Lincei Mat. Appl. 22, 89–111 (2011)
Bereanu C., Mawhin J.: Existence and multiplicity results for some nonlinear problems with singular \({\phi}\)-laplacian. J. Differential Equations 243, 536–557 (2007)
Bereanu C., Mawhin J.: Non-homogeneous boundary value problems for some nonlinear equations with singular \({\phi}\)-laplacian. J. Math. Anal. Appl. 352, 218–233 (2009)
C. Bereanu and P.J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., to appear.
Brezis H., Mawhin J.: Periodic solutions of the forced relativistic pendulum. Differential Integral Equations 23, 801–810 (2010)
Cid J.A., Torres P.J.: Solvability for some boundary value problems with \({\phi}\)-Laplacian operators. Discrete Continuous Dynamical Systems-A 23, 727–732 (2009)
Chu J., Lei J., Zhang M.: The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator. J. Differential Equations 247, 530–542 (2009)
Deimling K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Ferracuti L., Papalini F.: Boundary-value problems for strongly nonlinear multivalued equations involving different Φ-Laplacians. Adv. Differential Equations 14, 541–566 (2009)
Jebelean P.: Variational methods for ordinary p-Laplacian systems with potential boundary conditions. Adv. Differential Equations 14, 273–322 (2008)
J. Mawhin, Global results for the forced pendulum equation, in Handbook of Differential Equations. Ordinary Differential Equations, Vol. 1, A. Cañada, P. Drábek, A. Fonda ed., Elsevier, Amsterdam, 2004, 533–590.
J. Mawhin, Boundary value problems for nonlinear perturbations of singular \({\phi}\)-Laplacians, in Differential equations, chaos and variational problems (Aveiro 2007), Progr. Nonlin. Diff. Eq. Appl., Vol. 75, Birkhauser, Basel, 2007, 247–256.
J. Mawhin, Multiplicity of solutions of variational systems involving \({\phi}\)-Laplacians with singular \({\phi}\) and periodic nonlinearities, submitted.
Mawhin J., Willem M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differential Equations 52, 264–287 (1984)
Serrin J.: On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 101, 139–167 (1961)
Szulkin A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 77–109 (1986)
Szulkin A.: A relative category and applications to critical point theory for strongly indefinite functionals. Nonlinear Anal. 15, 725–739 (1990)
Torres P.J.: Periodic oscillations of the relativistic pendulum with friction. Physics Letters A 372, 6386–6387 (2008)
Torres P.J.: Nondegeneracy of the periodically forced Liénard differential equation with \({\phi}\)-Laplacian. Comm. Contemporary Math. 13, 283–292 (2011)
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Mawhin, J. Radial Solutions of Neumann Problem for Periodic Perturbations of the Mean Extrinsic Curvature Operator. Milan J. Math. 79, 95–112 (2011). https://doi.org/10.1007/s00032-011-0148-5
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DOI: https://doi.org/10.1007/s00032-011-0148-5