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Transactions of the American Mathematical Society

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On the nonlinear eigenvalue problem $ \Delta u+\lambda e\sp u=0$


Authors: Takashi Suzuki and Ken’ichi Nagasaki
Journal: Trans. Amer. Math. Soc. 309 (1988), 591-608
MSC: Primary 35J65; Secondary 35P30, 47H12, 47H15
DOI: https://doi.org/10.1090/S0002-9947-1988-0961602-6
MathSciNet review: 961602
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Abstract: The structure of the set $ \mathcal{C}$ of solutions of the nonlinear eigenvalue problem $ \Delta u + \lambda {e^u} = 0$ under Dirichlet condition in a simply connected bounded domain $ \Omega $ is studied. Through the idea of parametrizing the solutions $ (u,\,\lambda )$ in terms of $ s = \lambda \,\int_\Omega {{e^u}\,dx} $, some profile of $ \mathcal{C}$ is illustrated when $ \Omega $ is star-shaped. Finally, the connectivity of the branch of Weston-Moseley's large solutions to that of minimal ones is discussed.


References [Enhancements On Off] (What's this?)

  • [1] C. Bandle, Existence theorems, qualitative results and a priori bounds for a class of a nonlinear Dirichlet problems, Arch. Rational Mech. Anal. 58 (1975), 219-238. MR 0454336 (56:12587)
  • [2] -, Isoperimetric inequalities for a nonlinear eigenvalue problem, Proc. Amer. Math. Soc. 56 (1976), 243-246. MR 0477402 (57:16930)
  • [3] -, Isoperimetric inequalities and applications, Pitman, Boston, Mass., 1980.
  • [4] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), 207-218. MR 0382848 (52:3730)
  • [5] H. Fujita, On the nonlinear equations $ \Delta u + {e^u} = 0$ and $ \partial v/\partial t = \Delta v + {e^v}$, Bull. Amer. Math. Soc. 75 (1969), 132-135. MR 0239258 (39:615)
  • [6] I. M. Gel'fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295-381. MR 0153960 (27:3921)
  • [7] B. Gidas, Wei-Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 544879 (80h:35043)
  • [8] H. R. Haegi, Extremalprobleme und Ungleichungen Konformer Gebietsgrössen, Compositio Math. 8 (1951), 81-111. MR 0039811 (12:602b)
  • [9] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241-269. MR 0340701 (49:5452)
  • [10] B. Kawohl, A remark on M. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem, Math. Methods Appl. Sci. 8 (1986), 93-101. MR 833253 (87h:35038)
  • [11] J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue problem, J. Differential Equations 16 (1974), 103-125. MR 0346305 (49:11030)
  • [12] H. B. Keller and D. S. Cohen, Some positive problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361-1376. MR 0213694 (35:4552)
  • [13] T. Laetsch, On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl. 35 (1971), 389-404. MR 0280869 (43:6588)
  • [14] J. Liouville, Sur l'équation aux différences partielles $ ({\partial ^2}\log \lambda )/\partial u\partial v \pm \lambda /2{a^2} = 0$, J. Math. 18 (1853), 71-72.
  • [15] J. L. Moseley, On asymptotic solutions for a Dirichlet problem with an exponential nonlinearity, Applied Math. Report no. 1, West Virginia Univ., 1981.
  • [16] -, Asymptotic solutions for a Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal. 14 (1983), 719-735. MR 704487 (84h:35062)
  • [17] K. Nagasaki and T. Suzuki, On a nonlinear eigenvalue problem, Lecture Notes in Numerical and Applied Analysis (K. Masuda and T. Suzuki, eds.), Kinokuniya-North Holland, Tokyo and Amsterdam, 1987. MR 928192 (89j:35055)
  • [18] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 0045823 (13:640h)
  • [19] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202. MR 0320850 (47:9383)
  • [20] S. Richardson, Vortices, Liouville's equation and the Bergman kernel equation, Matematika 27 (1980), 321-334. MR 610715 (83m:76026)
  • [21] H. Wente, Counter example to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-244.
  • [22] V. H. Weston, On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal. 9 (1978), 1030-1053. MR 512508 (80a:35022)
  • [23] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161-180. MR 0341212 (49:5962)

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DOI: https://doi.org/10.1090/S0002-9947-1988-0961602-6
Article copyright: © Copyright 1988 American Mathematical Society

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