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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 5, pp 1585–1611 | Cite as

Radially symmetric solutions of elliptic PDEs with uniformly negative weight

  • Christopher S. Goodrich
Article
  • 70 Downloads

Abstract

We consider the perturbed Hammerstein integral equation
$$\begin{aligned} y(t)=\gamma (t)H(\varphi (y))+\lambda \int _0^1G(t,s)f(s,y(s))\, \mathrm{d}s \end{aligned}$$
in the case where it may hold that \(f(t,y)<0\), for each \((t,y)\in [0,1]\times [0,+\infty )\), and \(\lim _{y\rightarrow \infty }f(t,y)=-\infty \); in other words, f may be a strictly negative function on its entire domain and may uniformly blow up to \(-\infty \) as \(y\rightarrow +\infty \). We apply our results, in part, to radially symmetric solutions of PDEs of the form
$$\begin{aligned} -\Delta u(\varvec{x})=\lambda a(|\varvec{x}|)g(u(\varvec{x})) \end{aligned}$$
subject to nonlocal boundary conditions and show that this problem can possess a positive solution even if \(\lim _{u\rightarrow \infty }g(u)=-\infty \). By using a nonstandard cone and attendant open set, these results are able to be guaranteed by imposing relatively straightforward conditions. In addition, our results apply to forcing terms f and g with polynomial growth at \(+\infty \) of any degree. We demonstrate that, in principle, our results can be applied to ecological modeling with density-dependent growth and nonlocal boundary conditions.

Keywords

Hammerstein integral equation Negative weight Coercivity Radially symmetric solution Density-dependent growth 

Mathematics Subject Classification

Primary 35B09 35J25 45G10 45M20 47H30 Secondary 34B10 47H07 92D40 

References

  1. 1.
    Anderson, D.R., Zhai, C.: Positive solutions to semi-positone second-order three-point problems on time scales. Appl. Math. Comput. 15, 3713–3720 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, D.R.: Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions. J. Math. Anal. Appl. 408, 318–323 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anuradha, V., Hai, D.D., Shivaji, R.: Existence results for superlinear semipositone BVPs. Proc. Am. Math. Soc. 124, 757–763 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabada, A., Infante, G., Tojo, F.: Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications. Topol. Methods Nonlinear Anal. 47, 265–287 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cabada, A., Infante, G., Tojo, F.A.F.: Nonlinear perturbed integral equations related to nonlocal boundary value problems. Fixed Point Theory 19, 65–92 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cianciaruso, F., Pietramala, P.: Multiple positive solutions of a \((p_1, p_2)\)-Laplacian system with nonlinear BCs. Bound. Value Probl. 2015, 163 (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cianciaruso, F., Infante, G., Pietramala, P.: Solutions of perturbed Hammerstein integral equations with applications. Nonlinear Anal. Real World Appl. 33, 317–347 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahal, R.: Positive solutions of semipositone singular Dirichlet dynamic boundary value problems. Nonlinear Dyn. Syst. Theory 9, 361–374 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dhanya, R., Morris, Q., Shivaji, R.: Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball. J. Math. Anal. Appl. 434, 1533–1548 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    do Ó, J.M., Lorca, S., Ubilla, P.: Local superlinearity for elliptic systems involving parameters. J. Differ. Equ. 211, 1–19 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    do Ó, J.M., Lorca, S., Ubilla, P.: Three positive solutions for a class of elliptic systems in annular domains. Proc. Edinb. Math. Soc. 48, 365–373 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    do Ó, J.M., Lorca, S., Ubilla, P.: Positive solutions for a class of multi parameter ordinary elliptic systems. J. Math. Anal. Appl. 332, 1249–1266 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Feltrin, G., Zanolin, F.: Multiple positive solutions for a superlinear problem: a topological approach. J. Differ. Equ. 259, 925–963 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goodrich, C.S.: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 75, 417–432 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Goodrich, C.S.: On nonlocal BVPs with boundary conditions with asymptotically sublinear or superlinear growth. Math. Nachr. 285, 1404–1421 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Goodrich, C.S.: On nonlinear boundary conditions satisfying certain asymptotic behavior. Nonlinear Anal. 76, 58–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goodrich, C.S.: On a nonlocal BVP with nonlinear boundary conditions. Results Math. 63, 1351–1364 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goodrich, C.S.: A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions. Arch. Math. (Basel) 103, 177–187 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Goodrich, C.S.: Semipositone boundary value problems with nonlocal, nonlinear boundary conditions. Adv. Differ. Equ. 20, 117–142 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Goodrich, C.S.: On nonlinear boundary conditions involving decomposable linear functionals. Proc. Edinb. Math. Soc. (2) 58, 421–439 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Goodrich, C.S.: Coercivity of linear functionals on finite dimensional spaces and its application to discrete BVPs. J. Differ. Equ. Appl. 22, 623–636 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goodrich, C.S.: Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs. J. Integral Equ. Appl. 28, 509–549 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goodrich, C.S.: Coercive nonlocal elements in fractional differential equations. Positivity 21, 377–394 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Goodrich, C.S.: A new coercivity condition applied to semipositone integral equations with nonpositive, unbounded nonlinearities and applications to nonlocal BVPs. J. Fixed Point Theory Appl. 19, 1905–1938 (2017).  https://doi.org/10.1007/s11784-016-0340-x MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Goodrich, C.S.: On semipositone nonlocal boundary value problems with nonlinear or affine boundary conditions. Proc. Edinb. Math. Soc. 60, 635–649 (2017).  https://doi.org/10.1017/S0013091516000146 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Goodrich, C.S.: The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems. J. Fixed Point Theory Appl. 19, 2629–2646 (2017).  https://doi.org/10.1007/s11784-017-0448-7 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Goodrich, C.S.: New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green’s function. J. Differ. Equ. 264, 236–262 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Graef, J., Kong, L., Wang, H.: A periodic boundary value problem with vanishing Green’s function. Appl. Math. Lett. 21, 176–180 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Graef, J., Kong, L.: Positive solutions for third order semipositone boundary value problems. Appl. Math. Lett. 22, 1154–1160 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Graef, J., Webb, J.R.L.: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Anal. 71, 1542–1551 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)zbMATHGoogle Scholar
  32. 32.
    Herrón, S., Lopera, E.: Non-existence of positive radial solution for semipositone weighted \(p\)-Laplacian problem. Electron. J. Differ. Equ. 2015, 1–9 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Infante, G.: Nonlocal boundary value problems with two nonlinear boundary conditions. Commun. Appl. Anal. 12, 279–288 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Infante, G., Pietramala, P.: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal. 71, 1301–1310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Infante, G., Pietramala, P.: Eigenvalues and non-negative solutions of a system with nonlocal BCs. Nonlinear Stud. 16, 187–196 (2009)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Infante, G., Pietramala, P.: Perturbed Hammerstein integral inclusions with solutions that change sign. Comment. Math. Univ. Carolin. 50, 591–605 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Infante, G., Pietramala, P.: A third order boundary value problem subject to nonlinear boundary conditions. Math. Bohem. 135, 113–121 (2010)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Infante, G., Minhós, F., Pietramala, P.: Non-negative solutions of systems of ODEs with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17, 4952–4960 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Infante, G., Pietramala, P.: Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions. Math. Methods Appl. Sci. 37, 2080–2090 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Infante, G., Pietramala, P., Tenuta, M.: Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory. Commun. Nonlinear Sci. Numer. Simul. 19, 2245–2251 (2014)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Infante, G., Pietramala, P.: Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains. Nonlinear Differ. Equ. Appl. NoDEA 22, 979–1003 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Jankowski, T.: Positive solutions to fractional differential equations involving Stieltjes integral conditions. Appl. Math. Comput. 241, 200–213 (2014)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Kang, S., Zhang, G., Shi, B.: Existence of three periodic positive solutions for a class of integral equations with parameters. J. Math. Anal. Appl. 323, 654–665 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Karakostas, G.L., Tsamatos, P. Ch.: Existence of multiple positive solutions for a nonlocal boundary value problem. Topol. Methods Nonlinear Anal. 19, 109–121 (2002)Google Scholar
  45. 45.
    Karakostas, G.L., Tsamatos, P. Ch.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002(30), 1–17 (2002)Google Scholar
  46. 46.
    Karakostas, G.L.: Existence of solutions for an \(n\)-dimensional operator equation and applications to BVPs. Electron. J. Differ. Equ. 71, 17 (2014)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Lan, K.Q.: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. (2) 63, 690–704 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Lan, K.Q.: Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems. Nonlinear Anal. 71, 5979–5993 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lan, K.Q.: Positive solutions of systems of Hammerstein integral equations. Commun. Appl. Anal. 15, 521–528 (2011)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Lan, K.Q., Yang, G.: A fixed point Index theory for nowhere normal-outward compact maps and applications. J. Appl. Anal. Comput. 6, 665–683 (2016)MathSciNetGoogle Scholar
  51. 51.
    Lan, K.Q., Lin, W.: Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations. J. Lond. Math. Soc. (2) 83, 449–469 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Lan, K.Q., Lin, W.: Positive solutions of systems of singular Hammerstein integral equations with applications to semi linear elliptic equations in annuli. Nonlinear Anal. 74, 7184–7197 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Lan, K.Q., Webb, J.R.L.: Positive solutions of semilinear differential equations with singularities. J. Differ. Equ. 148, 407–421 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Liu, X., Wu, J.: Positive solutions for a Hammerstein integral equation with a parameter. Appl. Math. Lett. 22, 490–494 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Ludwig, D., Aronson, D.C., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Ma, R., Zhong, C.: Existence of positive solutions for integral equations with vanishing kernels. Commun. Appl. Anal. 15, 529–538 (2011)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Picone, M.: Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 1–95 (1908)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Sun, J.P., Li, W.T.: Existence of positive solutions to semipositone Dirichlet BVPs on time scales. Dyn. Syst. Appl. 16, 571–578 (2007)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. NoDEA 15, 45–67 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. (2) 74, 673–693 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Webb, J.R.L., Infante, G.: Non-local boundary value problems of arbitrary order. J. Lond. Math. Soc. (2) 79, 238–258 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Webb, J.R.L., Infante, G.: Semi-positone nonlocal boundary value problems of arbitrary order. Commun. Pure Appl. Anal. 9, 563–581 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Webb, J.R.L.: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 4319–4332 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Webb, J.R.L.: Boundary value problems with vanishing Green’s function. Commun. Appl. Anal. 13, 587–595 (2009)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Webb, J.R.L.: Remarks on a non-local boundary value problem. Nonlinear Anal. 72, 1075–1077 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Whyburn, W.M.: Differential equations with general boundary conditions. Bull. Am. Math. Soc. 48, 692–704 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Xu, J., Yang, Z.: Positive solutions for a system of nonlinear Hammerstein integral equations and applications. J. Integral Equ. Appl. 24, 131–147 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Yang, Z.: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal. 62, 1251–1265 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Yang, Z.: Positive solutions of a second-order integral boundary value problem. J. Math. Anal. Appl. 321, 751–765 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Yang, Z.: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal. 65, 1489–1511 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Yang, Z.: Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions. Nonlinear Anal. 68, 216–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Yang, Z.: Positive solutions for a system of nonlinear Hammerstein integral equations and applications. Appl. Math. Comput. 218, 11138–11150 (2012)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  74. 74.
    Zhao, Z.: Positive solutions of semi-positone Hammerstein integral equations and applications. Appl. Math. Comput. 219, 2789–2797 (2012)MathSciNetzbMATHGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCreighton Preparatory SchoolOmahaUSA

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