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Advanced Nonlinear Studies

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Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

Julian López-Gómez
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  • Departamento de Análisis Matemático y Matemática Aplicada, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
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/ Pierpaolo OmariORCID iD: https://orcid.org/0000-0002-3601-7627
Published Online: 2019-05-16 | DOI: https://doi.org/10.1515/ans-2019-2048


This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem

{-(u1+u2)=λa(x)f(u)in (0,1),u(0)=0,u(1)=0,

where λ is a parameter, aL(0,1) changes sign, and fC1() is positive in (0,+). The attention is focused on the case f(0)=0 and f(0)=1, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

Keywords: Quasilinear Elliptic Equation; Prescribed Curvature Equation; Indefinite Problem; Neumann Condition; Bounded Variation Function; Positive Solution; Bifurcation; Connected Component

MSC 2010: 35J93; 34B18; 35B32


  • [1]

    S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439–475. CrossrefGoogle Scholar

  • [2]

    S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal. 141 (1996), no. 1, 159–215. CrossrefGoogle Scholar

  • [3]

    H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), no. 2, 336–374. CrossrefGoogle Scholar

  • [4]

    L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, New York, 2000. Google Scholar

  • [5]

    G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318. CrossrefGoogle Scholar

  • [6]

    G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc. 290 (1985), no. 2, 483–501. CrossrefGoogle Scholar

  • [7]

    G. Anzellotti, BV solutions of quasilinear PDEs in divergence form, Comm. Partial Differential Equations 12 (1987), no. 1, 77–122. CrossrefGoogle Scholar

  • [8]

    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78. CrossrefGoogle Scholar

  • [9]

    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572. CrossrefGoogle Scholar

  • [10]

    E. Bombieri, E. De Giorgi and M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Ration. Mech. Anal. 32 (1969), 255–267. CrossrefGoogle Scholar

  • [11]

    D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste 39 (2007), 63–85. Google Scholar

  • [12]

    D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations 243 (2007), no. 2, 208–237. CrossrefGoogle Scholar

  • [13]

    H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar

  • [14]

    K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl. 75 (1980), no. 1, 112–120. CrossrefGoogle Scholar

  • [15]

    M. Burns and M. Grinfeld, On a bistable quasilinear parabolic equation: Well-posedness and stationary solutions, Commun. Appl. Anal. 15 (2011), no. 2–4, 251–264. Google Scholar

  • [16]

    M. Burns and M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux, European J. Appl. Math. 22 (2011), no. 4, 317–331. CrossrefGoogle Scholar

  • [17]

    G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Oxford Lecture Ser. Math. Appl. 15, The Clarendon Press, New York, 1998. Google Scholar

  • [18]

    S. Cano-Casanova, J. López-Gómez and K. Takimoto, A quasilinear parabolic perturbation of the linear heat equation, J. Differential Equations 252 (2012), no. 1, 323–343. CrossrefGoogle Scholar

  • [19]

    S. Cano-Casanova, J. López-Gómez and K. Takimoto, A weighted quasilinear equation related to the mean curvature operator, Nonlinear Anal. 75 (2012), no. 15, 5905–5923. CrossrefGoogle Scholar

  • [20]

    P. Concus and R. Finn, On a class of capillary surfaces, J. Anal. Math. 23 (1970), 65–70. CrossrefGoogle Scholar

  • [21]

    C. Corsato, C. De Coster, N. Flora and P. Omari, Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry, Nonlinear Anal. 181 (2019), 9–23. CrossrefGoogle Scholar

  • [22]

    C. Corsato, C. De Coster and P. Omari, The Dirichlet problem for a prescribed anisotropic mean curvature equation: Existence, uniqueness and regularity of solutions, J. Differential Equations 260 (2016), no. 5, 4572–4618. CrossrefGoogle Scholar

  • [23]

    C. Corsato, P. Omari and F. Zanolin, Subharmonic solutions of the prescribed curvature equation, Commun. Contemp. Math. 18 (2016), no. 3, Article ID 1550042. Google Scholar

  • [24]

    L. Dascal, S. Kamin and N. A. Sochen, A variational inequality for discontinuous solutions of degenerate parabolic equations, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 99 (2005), no. 2, 243–256. Google Scholar

  • [25]

    M. Emmer, Esistenza, unicità e regolarità nelle superfici de equilibrio nei capillari, Ann. Univ. Ferrara Sez. VII (N. S.) 18 (1973), 79–94. Google Scholar

  • [26]

    D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Inst. Fundam. Res. Lect. Math. Phys. 81, Springer, Berlin, 1989. Google Scholar

  • [27]

    R. Finn, The sessile liquid drop. I. Symmetric case, Pacific J. Math. 88 (1980), no. 2, 541–587. CrossrefGoogle Scholar

  • [28]

    R. Finn, Equilibrium Capillary Surfaces, Grundlehren Math. Wiss. 284, Springer, New York, 1986. Google Scholar

  • [29]

    C. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature, J. Math. Pures Appl. (9) 58 (1979), no. 1, 75–109. Google Scholar

  • [30]

    C. Gerhardt, Global C1,1-regularity for solutions of quasilinear variational inequalities, Arch. Ration. Mech. Anal. 89 (1985), no. 1, 83–92. Google Scholar

  • [31]

    E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (1976), no. 3, 501–548. Google Scholar

  • [32]

    E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984. Google Scholar

  • [33]

    R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations 167 (2000), no. 1, 36–72. CrossrefGoogle Scholar

  • [34]

    E. Gonzalez, U. Massari and I. Tamanini, Existence and regularity for the problem of a pendent liquid drop, Pacific J. Math. 88 (1980), no. 2, 399–420. CrossrefGoogle Scholar

  • [35]

    G. Huisken, Capillary surfaces over obstacles, Pacific J. Math. 117 (1985), no. 1, 121–141. CrossrefGoogle Scholar

  • [36]

    A. Kurganov and P. Rosenau, On reaction processes with saturating diffusion, Nonlinearity 19 (2006), no. 1, 171–193. CrossrefGoogle Scholar

  • [37]

    O. A. Ladyzhenskaya and N. N. Ural’tseva, Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pure Appl. Math. 23 (1970), 677–703. CrossrefGoogle Scholar

  • [38]

    V. K. Le, Some existence results on nontrivial solutions of the prescribed mean curvature equation, Adv. Nonlinear Stud. 5 (2005), no. 2, 133–161. Google Scholar

  • [39]

    V. K. Le and K. Schmitt, Global Bifurcation in Variational Inequalities. Applications to Obstacle and Unilateral Problems, Appl. Math. Sci. 123, Springer, New York, 1997. Google Scholar

  • [40]

    J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Res. Notes Math. 426, Chapman & Hall/CRC, Boca Raton, 2001. Google Scholar

  • [41]

    J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn. 61 (2005), no. 3, 493–516. Google Scholar

  • [42]

    J. López-Gómez and P. Omari, Positive solutions of a sublinear indefinite quasilinear Neumann problem, in preparation.

  • [43]

    J. López-Gómez, P. Omari and S. Rivetti, Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal. 155 (2017), 1–51. CrossrefGoogle Scholar

  • [44]

    J. López-Gómez, P. Omari and S. Rivetti, Positive solutions of a one-dimensional indefinite capillarity-type problem: A variational approach, J. Differential Equations 262 (2017), no. 3, 2335–2392. CrossrefGoogle Scholar

  • [45]

    J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal. 13 (2014), no. 1, 1–73. Google Scholar

  • [46]

    M. Marzocchi, Multiple solutions of quasilinear equations involving an area-type term, J. Math. Anal. Appl. 196 (1995), no. 3, 1093–1104. CrossrefGoogle Scholar

  • [47]

    M. Nakao, A bifurcation problem for a quasi-linear elliptic boundary value problem, Nonlinear Anal. 14 (1990), no. 3, 251–262. CrossrefGoogle Scholar

  • [48]

    F. Obersnel and P. Omari, Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions, Differential Integral Equations 22 (2009), no. 9–10, 853–880. Google Scholar

  • [49]

    F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differential Equations 249 (2010), no. 7, 1674–1725. CrossrefGoogle Scholar

  • [50]

    F. Obersnel and P. Omari, Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation, Discrete Contin. Dyn. Syst. 33 (2013), no. 1, 305–320. Google Scholar

  • [51]

    F. Obersnel, P. Omari and S. Rivetti, Asymmetric Poincaré inequalities and solvability of capillarity problems, J. Funct. Anal. 267 (2014), no. 3, 842–900. CrossrefGoogle Scholar

  • [52]

    P. Rosenau, Free energy functionals at the high gradient limit, Phys. Rev. A 41 (1990), 2227–2230. CrossrefGoogle Scholar

  • [53]

    J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496. CrossrefGoogle Scholar

  • [54]

    R. Temam, Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Ration. Mech. Anal. 44 (1971/72), 121–156. CrossrefGoogle Scholar

About the article

Received: 2019-03-11

Accepted: 2019-04-26

Published Online: 2019-05-16

Funding Source:

The authors have been supported by “Università degli Studi di Trieste-Finanziamento di Ateneo per Progetti di Ricerca Scientifica-FRA 2015” and by the INdAM-GNAMPA 2017 Research Project “Problemi fortemente nonlineari: esistenza, molteplicità, regolarità delle soluzioni”. This paper has been written under the auspices of the Ministry of Science, Technology and Universities of Spain under Research Grants MTM2015-65899-P and PGC2018-097104-B-100.

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2019-2048.

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