Symmetric positive solutions to a singular φ-Laplace equation


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.


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

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



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P. Jebelean, R. Precup, Symmetric positive solutions to a singular φ-Laplace equation, J. London Math. Soc.  99 (2019), 495-515,



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[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).

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