## Abstract

The localization of positive symmetric solutions to the Dirichlet problem for second-order ordinary differential equations involving a singular φ-Laplacian is established in a conical annular set, via Ekeland’s variational principle, compression type conditions, and a Harnack type inequality. An application to a one-parameter problem is provided and multiple such solutions are obtained in the case of oscillatory nonlinearities.

## Authors

**Petru Jebelean
**Department of Mathematics, West University of Timişoara, Timişoara, Romania

**Radu Precup**

Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

?

## Paper coordinates

P. Jebelean, R. Precup, *Symmetric positive solutions to a singular φ-Laplace equation*, J. London Math. Soc. 99 (2019), 495-515, https://doi.org/10.1112/jlms.12183

??

## About this paper

##### Journal

Journal London Mathematical Society

##### Publisher Name

London Mathematical Society

##### Print ISSN

?

##### Online ISSN

1469-7750

google scholar link

[1] R. Bartnik and L. Simon, ‘Spacelike hypersurfaces with prescribed boundary values and mean curvature’, Comm. Math. Phys. 87 (1982–83) 131–152.

[2] C. Bereanu, P. Jebelean and J. Mawhin, ‘Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces’, Proc. Amer. Math. Soc. 137 (2009) 161–169.

[3] C. Bereanu and J. Mawhin, ‘Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian’, J. Differential Equations 243 (2007) 536–557.

[4] C. Bereanu and J. Mawhin, ‘Boundary value problems for some nonlinear systems with singular φ-laplacian’, J. Fixed Point Theory Appl. 4 (2008) 57–75.

[5] C. Bereanu and J. Mawhin, ‘Nonhomogeneous boundary value problems for some nonlinear equations with singular φ-Laplacian’, J. Math. Anal. Appl. 352 (2009) 218–233.

[6] I. Coelho, C. Corsato, F. Obersnel and P. Omari, ‘Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation’, Adv. Nonlinear Stud. 12 (2012) 621–638.

[7] C. De Coster and P. Habets, Two-point boundary value problems: lower and upper solutions (Elsevier, Amsterdam, 2006).

[8] C. Gerhardt, ‘H-surfaces in Lorentzian manifolds’, Comm. Math. Phys. 89 (1983) 523–553.

[9] A. Granas and J. Dugundji, Fixed point theory (Springer, New York, 2003).

[10] D.-R. Herlea, ‘Positive solutions for second-order boundary-value problems with φ-Laplacian’, Electron. J. Differential Equations 2016 (2016) 1–8.

[11] P. Jebelean, J. Mawhin and C. S¸erban, ‘Multiple periodic solutions for perturbed relativistic pendulum systems’, Proc. Amer Math. Soc. 143 (2015) 3029–3039.

[12] P. Jebelean and C. S¸erban, ‘Boundary value problems for discontinuous perturbations of singular φ-Laplacian operator’, J. Math. Anal. Appl. 431 (2015) 662–681.

[13] J. L. W. V. Jensen, ‘Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes’, Acta Math. 30 (1906) 175–193.

[14] M. A. Krasnoselskii, Positive solutions of operator equations (Noordhoff, Groningen, 1964).

[15] H. Lisei, R. Precup and C. Varga, ‘A Schechter type critical point result in annular conical domains of a Banach space and applications’, Discrete Contin. Dyn. Syst. 36 (2016) 3775–3789.

[16] A. Capietto, P. Kloeden, J. Mawhin, S. Novo and M. Ortega ‘Resonance problems for some non-autonomous ordinary differential equations’, Stability and bifurcation theory for non-autonomous differential equations, Lecture Notes in Mathematics 2065 (eds R. Johnson and M. P. Pera; Springer, Berlin, Heidelberg, 2013) 103–184.

[17] R. Precup, ‘Positive solutions of semi-linear elliptic problems via Krasnoselskii type theorems in cones and Harnack’s inequality’, AIP Conf. Proc. 835 (2006) 125–132.

[18] R. Precup, ‘Critical point theorems in cones and multiple positive solutions of elliptic problems’, Nonlinear Anal. 75 (2012) 834–851.

[19] R. Precup, ‘Critical point localization theorems via Ekeland’s variational principle’, Dynam. Systems Appl. 22 (2013) 355–370.

[20] A. W. Roberts and D. E. Varberg, Convex functions (Academic Press, New York, 1973).

[21] M. Schechter, Linking methods in critical point theory (Birkh¨auser, Boston, 1999).