zbMATH — the first resource for mathematics

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities. (English) Zbl 1262.35088
Summary: We show that if \({{\mathcal A} \subset \mathbb{R}^N}\) is an annulus or a ball centered at zero, the homogeneous Neumann problem on \({{\mathcal A}}\) for the equation with continuous data \[ \nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|) \] has at least one radial solution when \(g(|x|,\cdot )\) has a periodic indefinite integral and \({\int_{\mathcal A} h(|x|)\,{\mathrm{d}}x = 0.}\) The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J93 Quasilinear elliptic equations with mean curvature operator
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
Full Text: DOI
[1] Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152 (1982/83) · Zbl 0512.53055
[2] Benevieri P., do Ó J.M., de Souto E.M.: Periodic solutions for nonlinear systems with mean curvature-like operators. Nonlinear Anal. 65, 1462–1475 (2006) · Zbl 1106.34024
[3] Benevieri P., do Ó J.M., de Souto E.M.: Periodic solutions for nonlinear equations with mean curvature-like operators. Appl. Math. Lett. 20, 484–492 (2007) · Zbl 1146.34034
[4] Bereanu C., Jebelean P., Mawhin J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidian and Minkowski spaces. Proc. Am. Math. Soc. 137, 161–169 (2009) · Zbl 1161.35024
[5] Bereanu C., Jebelean P., Mawhin J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowsi spaces. Math. Nachr. 283, 379–391 (2010) · Zbl 1185.35113
[6] Bereanu C., Jebelean P., Mawhin J.: Radial solutions for Neumann problems with $${\(\backslash\)phi}$$ -Laplacians and pendulum-like nonlinearities. Discret. Cont. Dynam. Syst. A 28, 637–648 (2010) · Zbl 1193.35083
[7] Bereanu C., Mawhin J.: Nonlinear Neumann boundary value problems with $${\(\backslash\)phi}$$ -Laplacian operators. An. Stiint. Univ. Ovidius Constanta 12, 73–92 (2004) · Zbl 1117.34015
[8] Bereanu C., Mawhin J.: Periodic solutions of nonlinear perturbations of $${\(\backslash\)phi}$$ -Laplacian with possibly bounded $${\(\backslash\)phi}$$ . Nonlinear Anal. 68, 1668–1681 (2008) · Zbl 1147.34032
[9] Bonheure D., Habets P., Obersnel F., Omari P.: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 243, 208–237 (2007) · Zbl 1136.34023
[10] Brezis H., Mawhin J.: Periodic solutions of the forced relativistic pendulum. Differ. Integr. Equ. 23, 801–810 (2010) · Zbl 1240.34207
[11] Girg P.: Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivatives. Electron. J. Differ. Equ. 63, 1–28 (2000) · Zbl 0974.34018
[12] Habets P., Omari P.: Multiple positive solutions of a one-dimensional prescribed mean curvature problem. Commun. Contemp. Math. 9, 701–730 (2007) · Zbl 1153.34015
[13] Mawhin, J.: Global results for the forced pendulum equation. In: Cañada, A., Drábek, P., Fonda, A. (eds.) Handbook of Differential Equations. Ordinary Differential Equations, vol. 1, pp. 533–590. Elsevier, Amsterdam (2004) · Zbl 1091.34019
[14] Obersnel F., Omari P.: Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions. Differ. Integr. Equ. 22, 853–880 (2009) · Zbl 1240.35131
[15] Rachunková I., Staněk S., Tvrdý M.: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi Publishing Corporation, New York (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.