Radially symmetric solutions to a Dirichlet problem involving critical exponents
Authors:
Alfonso Castro and Alexandra Kurepa
Journal:
Trans. Amer. Math. Soc. 343 (1994), 907926
MSC:
Primary 35B05; Secondary 35J65
MathSciNet review:
1207581
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Abstract: In this paper we answer, for , the question raised in [1] on the number of radially symmetric solutions to the boundary value problem , , , , where is the Laplacean operator and . Indeed, we prove that if , then for any this problem has only finitely many radial solutions. For we show that, for each , the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
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 [1]
 F. Atkinson, H. Brezis, and L. Peletier, Solutions d'equations elliptiques avec exposant de Sobolev critique qui changent de signe, C.R. Acad. Sci. Paris Ser. I Math. 306 (1988), 711714. MR 944417 (89k:35088)
 [2]
 H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437477. MR 709644 (84h:35059)
 [3]
 A. Castro and A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc. 101 (1987), 5664. MR 897070 (88j:35058)
 [4]
 A. Castro and A. Kurepa, Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities, Trans. Amer. Math. Soc. 315 (1989), 353372. MR 933323 (90g:35053)
 [5]
 G. Cerami, Elliptic equations with critical growth, College on Variational Problems in Analysis, Lecture Notes SMR 281/24, Internat. Centre for Theoretical Physics, Trieste, Italy, 1988.
 [6]
 G. Cerami, S. Solimini, and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), 289306. MR 867663 (88b:35074)
 [7]
 M. G. Crandall, and P. M. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321340. MR 0288640 (44:5836)
 [8]
 M. K. Kwong, Uniqueness of positive solutions for in , Arch. Rational Mech. Anal. 105 (1989), 243266. MR 969899 (90d:35015)
 [9]
 S. I. Pohozaev, Eigenfunctions of the equation , Soviet. Math. Dokl. 6 (1965), 14081411.
 [10]
 P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681703. MR 855181 (88b:35072)
 [11]
 M. Shinbrot and R. Welland, The CauchyKowalewskaya theorem, J. Math. Anal. Appl. 55 (1976), 757772. MR 0492756 (58:11827)
 [12]
 S. Solimini, On the existence of infinitely many radial solutions for some elliptic problems, Rev. Mat. Apl. 9 (1987), 7586. MR 926233 (89f:35086)
 [13]
 N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sci. Norm. Sup. Pisa 22 (1968), 265274. MR 0240748 (39:2093)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412075810
PII:
S 00029947(1994)12075810
Keywords:
Critical exponent,
radially symmetric solutions,
Dirichlet problem,
nodal curves,
bifurcation
Article copyright:
© Copyright 1994
American Mathematical Society
