Fractional Generalized Logistic Equations with Indefinite Weight: Quantitative and Geometric Properties

  • Alessio Marinelli
  • Dimitri MugnaiEmail author
Part of the following topical collections:
  1. Perspectives of Geometric Analysis in PDEs


We deal with fractional generalized logistic problems in presence of a signed and unbounded weight. We describe the first eigenpair of the underlying operators and we show a bifurcation result for positive solutions, which are proved to be unique. A symmetry result is established under suitable geometric constraints.


Fractional Laplacians Indefinite potential Principal eigenfunction Positive solutions Axial symmetry 

Mathematics Subject Classification

35J20 35J65 58E05 



  1. 1.
    Alama, S., Tarantello, G.: Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141, 159–215 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alves, M.O., Pimenta, M.T.O., Suárez, A.: Lotka-Volterra models with fractional diffusion. Proc. R. Soc. Edinb. A 147, 505–528 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.: The non-local Fisher-KPP equation: travelling waves and steady states. Nonlinearity 22, 2813–2844 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Berestycki, H., Roquejoffre, J.-M., Rossi, L.: The periodic patch model for population dynamics with fractional diffusion. Discret. Contin. Dyn. Syst. S 4, 1–13 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Balackrishnan, A.V.: Fractional powers of closed operators an the semigroups generated by them. Pac. J. Math. 10, 419–437 (1960)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brasco, L., Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. 37, 769–799 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20 (2016)Google Scholar
  9. 9.
    Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, 2052–2093 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cabré, X., Roquejoffre, J.M.: The Influence of Fractional Diffusion in Fisher-KPP Equations. Commun. Math. Phys. 320, 679–722 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32, 1245–1260 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Caffarelli, L., Vasseur, A.: Drift diffusion equation with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 2(171), 903–930 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Non-local minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)zbMATHGoogle Scholar
  14. 14.
    Caffarelli, L., Dipierro, S., Valdinoci, E.: A logistic equation with nonlocal interactions. Kinet. Relat. Models 10, 141–170 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Cantrell, R.S., Cosner, C., Hutson, V.: Ecological models, permanence and spatial heterogeneity. Rocky Mt. J. Math. 26, 1–35 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cantrell, R.S., Cosner, C., Hutson, V.: Spatial Ecology via Reaction-diffusion Equations. Wiley, New York (2003)zbMATHGoogle Scholar
  17. 17.
    Carboni, G., Mugnai, D.: On some fractional equations with convex-concave and logistic-type nonlinearities. J. Differ. Equ. 262, 2393–2413 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Costa, D.G., Drábek, P., Tehrani, H.T.: Positive solutions to semilinear elliptic equations with logistic type nonlinearities and constant yield harvesting in \(\mathbb{R}^N\). Commun. Partial Differ. Equ. 33, 1597–1610 (2008)zbMATHCrossRefGoogle Scholar
  19. 19.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Dipierro, S., Figalli, A., Valdinoci, E.: Strongly Nonlocal Dislocation Dynamics in Crystals. Commun. Partial Differ. Equ. 39, 2351–2387 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Du, Y., Ma, L.: Logistic type equations on \(\mathbb{R}^N\) by a squeezing method involving boundary blow-up solutions. J. Lond. Math. Soc. 2(64), 107–124 (2001)zbMATHCrossRefGoogle Scholar
  22. 22.
    Felmer, P., Wang, Y.: Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun. Contemp. Math. 16(01), 1350023 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schroedinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh A 142, 1237–1262 (2012)zbMATHCrossRefGoogle Scholar
  24. 24.
    Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)zbMATHCrossRefGoogle Scholar
  25. 25.
    Fragnelli, G., Mugnai, D., Papageorgiou, N.S.: Robin problems for the \(p\)-Laplacian with gradient dependence. Discret. Contin. Dyn. Syst. S 12, 287–295 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 315–328 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Girao, P., Tehrani, H.: Positive solutions to logistic type equations with harvesting. J. Differ. Equ. 247, 574–595 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Marinelli, A., Mugnai, D.: Generalized logistic equation with indefinite weight driven by the square root of the Laplacian. NonLinearity 27, 1–16 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Massaccesi, A., Valdinoci, E.: Is a nonlocal diffusion strategy convenient for biological populations in competition? J. Math. Biol. 74, 113–147 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Molica Bisci, G., Radulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2016)zbMATHCrossRefGoogle Scholar
  32. 32.
    Montefusco, E., Pellacci, B., Verzini, G.: Fractional diffusion with Neumann boundary conditions: the logistic equation. Discret. Contin. Dyn. Syst. B 18, 2175–2202 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Mugnai, D., Papageorgiou, N.: Bifurcation for positive solutions of nonlinear diffusive logistic equations in \(\mathbb{R}^N\) with indefinite weight. Indiana Univ. Math. J. 63, 1397–1418 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Neand, Y., Nepomnyashchy, A.A.: Turing instability of anomalous reaction-anomalous diffusion systems. Eur. J. Appl. Math. 19, 329–349 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Pelcé, P.: New Visions on Form and Growth. Oxford University, New York (2004)Google Scholar
  36. 36.
    Radulescu, V., Repovs, D.: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal. 75, 1524–1530 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ros-Oton, X., Serra, J.: The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50, 723–750 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ruiz, D., Suarez, A.: Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term. Proc. R. Soc. Edinb. A 137, 555–566 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discret. Contin. Dyn. Syst. 33, 2105–2137 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58, 133–154 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Shi, J., Shivaji, R.: Semilinear elliptic equations with generalized cubic nonlinearities. In: Proceedings of the Fifth International Conference on Dynamical Systems on Difference Equations, Pomona, CA (2005)Google Scholar
  43. 43.
    Song, R., Vondraček, Z.: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields 125, 578–592 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Szulkin, A., Willem, M.: Eigenvalue problems with indefinite weight. Studia Math. 135, 191–201 (1999)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Torres Ledesma, C.E.: Existence and symmetry result for fractional \(p-\)Laplacian in \(\mathbb{R}^N\). Commun. Pure Appl. Anal. 16, 99–113 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Vazquez, J.L.: Nonlinear diffusion with fractional laplacian operators. In: Holden, H., Karlsen, K.H. (eds.) Nonlinear Partial Differential Equations: The Abel Symposium 2010, pp. 271–298. Springer, New York (2012)CrossRefGoogle Scholar
  47. 47.
    Yang, R., Lü, Z.: The properties of positive solutions to semilinear equations involving the fractional Laplacian. Commun. Pure Appl. Anal. 18, 1073–1089 (2019)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly
  2. 2.Department of Ecology and Biology (DEB)Tuscia University, Largo dell’UniversitàViterboItaly

Personalised recommendations