Advertisement

Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on an Exterior Domain

  • Yongxiang Li
  • Yonghong Ding
  • Elyasa Ibrahim
Article
  • 60 Downloads

Abstract

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term:
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = K(|x|)\;f(|x|,\,u,\,|\nabla u|), \quad x\in \Omega ,\\ \alpha \,u+\beta \,\frac{\partial u}{\partial n}\;\big |_{\partial \Omega }=0,\\ \lim _{|x|\rightarrow \infty }u(x)=0, \end{array}\right. \end{aligned}$$
where \(\Omega =\{x\in \mathbb {R}^N:\;|x|>r_0\}\), \(N\ge 3\), \(K: [r_0,\,\infty )\rightarrow \mathbb {R}^+\) and \(f:[r_0,\,\infty )\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}^+\) are continuous, \(\mathbb {R}^+=[0,\,\infty )\). Under the assumption that the coefficient function K(r) satisfies \(0<\int _{r_0}^{\infty }r^{N-1}K(r)\,\mathrm{{d}}r<\infty \), and the conditions that the nonlinearity \(f(r,\,u,\,\eta )\) grows sub- or super-linear in u and \(\eta \), the existence results of positive radial solutions are obtained. For the superlinear case, the growth of f on \(\eta \) is restricted by a Nagumo-type condition and the coefficient function K(r) is further assumed to have the asymptotic behaviour that \(\;K(r)=O(1/r^{2(N-1)})\). The superlinear and sublinear growth of the nonlinearity f are described by inequality conditions instead of the usual upper and lower limits conditions. Our inequality conditions are weaker than the usual lower and upper limits conditions. The discussion is based on the fixed point index theory in cones.

Keywords

Elliptic equation positive radial solution exterior domain cone fixed point index 

Mathematics Subject Classification

35J25 35J60 47H11 47N20 

References

  1. 1.
    Butler, D., Ko, E., Shivaji, R.: Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete Cont. Dyn. Syst. Ser. S 7, 1181–1191 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Butler, D., Ko, E., Lee, E.K., Shivaji, R.: Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Commun. Pure Appl. Anal. 13, 2713–2731 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ehrnstrom, M.: On radial solutions of certain semi-linear elliptic equations. Nonlinear Anal. 64, 1578–1586 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erbe, L.H., Wang, H.: On the existence of the positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743–748 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D., Tanaka, M.: Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient. Nonlinear Anal. 96, 154–166 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)zbMATHGoogle Scholar
  8. 8.
    Ko, E., Ramaswamy, M., Shivaji, R.: Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball. J. Math. Anal. Appl. 423, 399–409 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lee, Y.H.: Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains. Differ. Integral Equ. 13, 631–648 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee, Y.H.: A multiplicity result ofpositive radial solutions for a multiparameter elliptic system on an exterior domain. Nonlinear Anal. 45, 597–611 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, Y.: Positive solutions of second order boundary value problems with sign-changing nonlinear terms. J. Math. Anal. Appl. 282, 232–240 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, Y.: On the existence and nonexistence of positive solutions for nonlinear Sturm-Liouville boundary value problems. J. Math. Anal. Appl. 304, 74–86 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, Y., Zhang, H.: Existence of positive radial solutions for the elliptic equations on an exterior domain. Ann. Pol. Math. 116, 67–78 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Liu, Z., Li, F.: Multiple positive solutions of nonlinear boundary problems. J. Math. Anal. Appl. 203, 610–625 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Precup, R.: Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems. J. Math. Anal. Appl. 352, 48–56 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ruiz, D.: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199, 96–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Santanilla, J.: Existence and nonexistence of positive radial solutions of an elliptic Dirichlet problem in an exterior domain. Nonlinear Anal. 25, 1391–1399 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Stanczy, R.: Decaying solutions for sublinear elliptic equations in exterior domains. Topol. Methods Nonlinear Anal. 14, 363–370 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stanczy, R.: Positive solutions for superlinear elliptic equations. J. Math. Anal. Appl. 283, 159–166 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tarfulea, N.: Positive radial symmetric solutions to an exterior elliptic Robin boundary-value problem and application. Nonlinear Anal. 71, 1909–1915 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vrdoljak, B.: Existence and behaviour of some radial solutions of a semilinear elliptic equation with a gradient-term. Math. Commun. 4, 11–17 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouPeople’s Republic of China

Personalised recommendations