On the eigenvalues of the Laplacian with varying
Author:
Yin Xi Huang
Journal:
Proc. Amer. Math. Soc. 125 (1997), 33473354
MSC (1991):
Primary 35P30, 35B30
MathSciNet review:
1403133
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the nonlinear eigenvalue problem where , is a bounded smooth domain in . We prove that the first and the second variational eigenvalues of (1) are continuous functions of . Moreover, we obtain the asymptotic behavior of the first eigenvalue as and .
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 [DEM]
 M. Del Pino, M. Elgueta and R. Manasevich, A homotopic deformation along of a LeraySchauder degree result and existence for , , , J. Diff. Equa. 80 (1989), 113. MR 91i:34018
 [D]
 J.I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries vol 1: Elliptic Equations, Research Notes in Mathematics 106, Pitman Advanced Publishing Program, London, 1985. MR 88d:35058
 [Eg]
 H. Egnell, Existence and nonexistence results for Laplace equations involving critical Sobolev exponents, Arch. Rat. Mech. Anal. 104 (1988), 5777. MR 90e:35069
 [GT]
 D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, SpringerVerlag, N.Y., 1983. MR 86c:35035
 [GV]
 M. Guedda and L. Veron, Bifurcation phenomena associated to the Laplace operator, Trans. Amer. Math. Soc. 310 (1988), 419431. MR 89j:35024
 [HM]
 Y.X. Huang and G. Metzen, The existence of solutions to a class of semilinear differential equations, Diff. Int. Equa. 8 (1995), 429452. MR 95h:34034
 [K1]
 B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. 410 (1990), 122. MR 91h:35126
 [K2]
 B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. # 1150, SpringerVerlag, N.Y. 1985. MR 87a:35001
 [LM]
 A.C. Lazer and P.J. McKenna, Largeamplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537578. MR 92g:73059
 [L1]
 P. Lindqvist, Stability for the solutions of with varying , J. Math. Anal. Appl. 127 (1987), 93102. MR 88i:35057
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 P. Lindqvist, On the equation , Proc. Amer. Math. Soc. 109 (1990), 157164. MR 90h:35088
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 [L4]
 P. Lindqvist, On a nonlinear eigenvalue problem: stability and concavity, Helsinki University of Technology, Inst. of Math. Research Reports # A279.
 [R]
 P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, AMS Regional Conference Series in Math. vol. 65 (1986). MR 87j:58024
 [Sz]
 A. Szulkin, LjusternikSchnirelmann theory on manifolds, Ann. Inst. Henri Poincaré, Anal. Nonl. 5 (1988), 119139. MR 90a:58027
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 P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. PDE 8 (1983), 773817.
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Additional Information
Yin Xi Huang
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email:
huangy@mathsci.msci.memphis.edu
DOI:
http://dx.doi.org/10.1090/S0002993997039610
PII:
S 00029939(97)039610
Keywords:
Eigenvalues,
the $p$Laplacian
Received by editor(s):
June 14, 1996
Additional Notes:
Research is partly supported by a University of Memphis Faculty Research Grant
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1997
American Mathematical Society
