• Chen, Tianlan (Department of Mathematics Northwest Normal University) ;
  • Lu, Yanqiong (Department of Mathematics Northwest Normal University) ;
  • Ma, Ruyun (Department of Mathematics Northwest Normal University)
  • Received : 2019.02.28
  • Accepted : 2019.07.08
  • Published : 2020.03.31


This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems -Δv = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ RN : r1 < |x| < r2} with 0 < r1 < r2, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s1(x)) = h(x, s2(x)) = 0 for suitable positive, concave function s1 and negative, convex function s2, as well as sh(x, s) > 0 for s ∈ ℝ \ {0, s1(x), s2(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.


Supported by : NSFC


  1. A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), no. 2, 411-422.
  2. G. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5323-5345.
  3. G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for p-Laplacian, J. Differential Equations 252 (2012), no. 3, 2448-2468.
  4. G. Dai, R. Ma, and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition, J. Math. Anal. Appl. 397 (2013), no. 1, 119-123.
  5. E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1973/74), 1069-1076. 23.23087
  6. D. G. de Figueiredo and J.-P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), no. 1-2, 339-346.
  7. D. D. Hai, Positive solutions for semilinear elliptic equations in annular domains, Non-linear Anal. 37 (1999), no. 8, Ser. A: Theory Methods, 1051-1058.
  8. L. Iturriaga, E. Massa, J. Sanchez, and P. Ubilla, Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros, Math. Nachr. 287 (2014), no. 10, 1131-1141.
  9. P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441-467.
  10. R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett. 21 (2008), no. 7, 754-760.
  11. R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal. 71 (2009), no. 10, 4364-4376.
  12. R. Ma and Y. An, Global structure of positive solutions for superlinear second order m-point boundary value problems, Topol. Methods Nonlinear Anal. 34 (2009), no. 2, 279-290.
  13. R. Ma, T. Chen, and Y. Lu, On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 8, 2649-2662.
  14. R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal. 59 (2004), no. 5, 707-718.
  15. R. Ma and B. Thompson, Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities, J. Math. Anal. Appl. 303 (2005), no. 2, 726-735.
  16. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, corrected reprint of the 1967 original, Springer-Verlag, New York, 1984.
  17. P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475.
  18. H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), no. 1, 1-7.
  19. G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23, Princeton University Press, Princeton, NJ, 1958.