East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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EXISTENCE OF A POSITIVE SOLUTION TO INFINITE SEMIPOSITONE PROBLEMS

  • Eunkyung Ko (Major in Mathematics, College of Natural Science, Keimyung University)
  • Received : 2024.02.02
  • Accepted : 2024.05.01
  • Published : 2024.05.31
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Abstract

We establish an existence result for a positive solution to the Schrödinger-type singular semipositone problem: $-{\Delta}u\,=\,V(x)u\,=\,{\lambda}{\frac{f(u)}{u^{\alpha}}}$ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN , N > 2, λ ∈ ℝ is a positive parameter, V ∈ L∞(Ω), 0 < α < 1, f ∈ C([0, ∞), ℝ) with f(0) < 0. In particular, when ${\frac{f(s)}{s^{\alpha}}}$ is sublinear at infinity, we establish the existence of a positive solutions for λ ≫ 1. The proofs are mainly based on the sub and supersolution method. Further, we extend our existence result to infinite semipositone problems with mixed boundary conditions.

Keywords

Acknowledgement

This work was financially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (NRF-2020R1F1A1A01065912).

References

  1. A. Ambrosetti, D. Arcoya and B. Biffoni, Positive solutions for some semipositone problems via bifurcation theory, Differential Integral Equations, 7(3) (1994), 655-663. 
  2. T. Ando, A. B. Fowler and F. Stern, Electronic properties of two-dimensional systems,Rev. Modern Phys. 54 (1982), no. 2, 437-672. 
  3. V. Anuradha, D. Hai and R. Shivaji, Existence results for superlinear semipositone boundary value problems, Proc. Amer. Math. Soc. 124(3) (1996),757-763. 
  4. T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrodinger equations, Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 191-198. 
  5. R. Bialecki and A. J. Nowak, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions , Appl. Math. Model. 5 (1981), no. 6, 417-421. 
  6. K. Brown, A. Castro and R. Shivaji, Nonexistence of radially symmetric solutions for a class of semipositone problems, Differential Integral Equations, 2(4) (1989), 541-545. 
  7. K.J. Brown, M.M.A. Ibrahim and R. Shivaji, S - shaphed bifurcation curves, Nonlinear Analysis, 5 (1981), 475-486. 
  8. A. Castro, D. G. de Figueredo and E. Lopera, Existence of positive solutions for a semipositone p-Laplacian problem, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 3, 475-482. 
  9. A. Castro, C. Maya and R. Shivaji, Nonlinear eigenvalue problems with semipositone structure, Electron. J. Differential Equations, Conf. 05 (2000), 33-49. 
  10. S. Cui, Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems, Nonlinear Analysis, 41 (2001),149-176. 
  11. J. M. Cushing, Nonlinear Steklov problems on the unit circle II (and a hydrodynamical application) , J. Math. Anal. Appl. 39 (1972), 267-278. 
  12. R. Dhanya, E. Ko and R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 424 (2015), 598-612. 
  13. D. Fasino and G. Inglese, Recovering unknown terms in a nonlinear boundary condition for Laplace's equation , IMA J. Appl. Math. 71 (2006), no. 6, 832-852 
  14. G.M. Figueiredo, J.R. Santos Junior and A. Suarez, Structure of the set of positive solutions of a non-linear Schrodinger equation, (English summary) Israel J. Math. 227 (2018), no. 1, 485-505. 
  15. P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion , Nonlinear Anal. Real World Appl. 15 (2014), 51-57. 
  16. Y. Guo, Z.-Q. Wang, X. Zeng and H.-S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity 31 (2018), no. 3, 957-979. 
  17. E. Ko, E.K. Lee and R. Shivaji, Multiplicity of positive solutions to a class of Schrodinger-typesingular problems, to appear in Discrete Contin. Dyn. Syst. Ser. S. 
  18. E. Ko, E.K. Lee and I. Sim, Existence of positive solution to Schrodinger-type semipositone problems with mixed nonlinear boundary conditions, Taiwanese J. Math. 25 (2021), no. 1, 107-124. 
  19. E. Ko and S. Prashanth, Positive solutions for elliptic equations in two dimensions arising in a theory of thermal explosion, Taiwanese J. Math. 19 (2015), no. 6, 1759-1775. 
  20. P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441-467. 
  21. J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrodinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), no. 5-6, 879-901. 
  22. M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of p-laplacian equations, Differential Integral Equations, 17 (2004), no. 11-12,1255 - 1261. 
  23. P.H. Rabinowitz, On a class of nonlinear Schrodinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270-291.