East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

PARAMETRIZED PERTURBATION RESULTS ON GLOBAL POSITIVE SOLUTIONS FOR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND HARDY TEREMS

  • Kim, Wan Se (Department of Mathematics Research Institute for Natural Sciences Hanyang University)
  • Received : 2017.12.11
  • Accepted : 2018.01.30
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We establish existence and bifurcation of global positive solutions for parametrized nonhomogeneous elliptic equations involving critical Sobolev-Hardy exponents and Hardy terms. The main approach to the problem is the variational method.

Keywords

References

  1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funt. Anal. 14, (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  2. H. Berestycki, L. Capffarelli and L. Nirenberg, Futher qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa CI. Sci. 425(1997),69-94.
  3. H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functions, Proc. Amer. Soc., 88 (1983), 486-490. https://doi.org/10.1090/S0002-9939-1983-0699419-3
  4. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36(1983), 437-477 . https://doi.org/10.1002/cpa.3160360405
  5. M. Crandall and P. Rabiowitz, Bifurcation, Perturbation of simple eigenvalues, and linealized stability, Arch. Rational Mech. Mech. Anal., 52(1973), 161-180. https://doi.org/10.1007/BF00282325
  6. L. Ekeland, Convex minimization problem, Bull. Amer. Math. Soc., (NS)1 (1976), 443-474.
  7. Y. Deng and Y. Li, Existence and bifurcation of positive solutions for a semilinear elliptic equation with critical exponent, J. Diff. Equa., 130 (1996), 179-200. https://doi.org/10.1006/jdeq.1996.0138
  8. N. Ghoussoub and C. Yuan, Multiple solutions for qusi-linear PDEs involving critical Sobolev and Hardy exponentials, Trans. Amer. Math. Soc., 352 (2000)5703-5743. https://doi.org/10.1090/S0002-9947-00-02560-5
  9. D. Kang, Solutions of quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term, J. Math. Anal. Appl., 341 (2008)464-781.
  10. D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev-Hardy exponentis, Nonlinear Analysis T.M.A., 60 (2005)297-753.
  11. D. Kang and S. Peng, Positive solutions for singular critical elliptic problems, Appl. Math. Lett., 17 (2004)411-416. https://doi.org/10.1016/S0893-9659(04)90082-1
  12. W.S. Kim, Multiple existence of positive global solutions parameterized nonhomogeneous elliptic equations involving critical exponents, East Asian Math. J., 30(3)(2014)335-353. https://doi.org/10.7858/eamj.2014.024
  13. D. Smets, Nonlinear Schrodinger equations with Hardy potentials and critical nonlearitie, Trans. Amer. Math. Soc., 357(7) (2004) 2909-2938. https://doi.org/10.1090/S0002-9947-04-03769-9
  14. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976). 353-372. https://doi.org/10.1007/BF02418013
  15. M. Willem, Minimax theorems, Birkhauser. Boston, Basel, Berlin. (1996).
  16. X.-P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Diff. Equa, 92 (1991) 163-178. https://doi.org/10.1016/0022-0396(91)90045-B
  17. X.-P. Zhu and H.-S. Zhu, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domain, Proc. Roy. Soc. Edinburgh , 115A(1990)301-318.