대한수학회지 (Journal of the Korean Mathematical Society)

  • 제47권3호
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  • Pages.547-572
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  • 2010
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
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EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

  • Zou, Henghui (DEPARTMENT OF MATHEMATICS UNIVERSITY OF ALABAMA AT BIRMINGHAM)
  • 발행 : 2010.05.01
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초록

We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

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참고문헌

  1. A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  2. T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.
  3. M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1-49. https://doi.org/10.1007/BF02788105
  4. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477. https://doi.org/10.1002/cpa.3160360405
  5. E. DiBenedetto, $C^{1+{\alpha}}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827-850. https://doi.org/10.1016/0362-546X(83)90061-5
  6. Y. Ding and F. Lin, Solutions of perturbed Schrodinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), no. 2, 231-249. https://doi.org/10.1007/s00526-007-0091-z
  7. P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201.
  8. P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121.
  9. P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), no. 3, 681-703. https://doi.org/10.1512/iumj.1986.35.35036
  10. P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
  11. J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302. https://doi.org/10.1007/BF02391014
  12. J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), no. 1, 79-142. https://doi.org/10.1007/BF02392645
  13. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. https://doi.org/10.1007/BF02418013
  14. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150. https://doi.org/10.1016/0022-0396(84)90105-0
  15. N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. https://doi.org/10.1002/cpa.3160200406
  16. J.-L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191-202. https://doi.org/10.1007/BF01449041
  17. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, Birkhauser Boston, Inc., Boston, MA, 1996.

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