On the eigenvalues of the -Laplacian

with varying

Author:
Yin Xi Huang

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3347-3354

MSC (1991):
Primary 35P30, 35B30

MathSciNet review:
1403133

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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 .

**[AA]**J.P. G. Azorero and I.P. Alonso,*Existence and uniqueness for the -Laplacian: nonlinear eigenvalues*, Comm. PDE 12 (1987), 1389-1430.**[DEM]**Manuel del Pino, Manuel Elgueta, and Raúl Manásevich,*A homotopic deformation along 𝑝 of a Leray-Schauder degree result and existence for (\vert𝑢’\vert^{𝑝-2}𝑢’)’+𝑓(𝑡,𝑢)=0,𝑢(0)=𝑢(𝑇)=0,𝑝>1*, J. Differential Equations**80**(1989), no. 1, 1–13. MR**1003248**, 10.1016/0022-0396(89)90093-4**[D]**J. I. Díaz,*Nonlinear partial differential equations and free boundaries. Vol. I*, Research Notes in Mathematics, vol. 106, Pitman (Advanced Publishing Program), Boston, MA, 1985. Elliptic equations. MR**853732****[Eg]**Henrik Egnell,*Existence and nonexistence results for 𝑚-Laplace equations involving critical Sobolev exponents*, Arch. Rational Mech. Anal.**104**(1988), no. 1, 57–77. MR**956567**, 10.1007/BF00256932**[GT]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[GV]**Mohammed Guedda and Laurent Véron,*Bifurcation phenomena associated to the 𝑝-Laplace operator*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 419–431. MR**965762**, 10.1090/S0002-9947-1988-0965762-2**[HM]**Yin Xi Huang and Gerhard Metzen,*The existence of solutions to a class of semilinear differential equations*, Differential Integral Equations**8**(1995), no. 2, 429–452. MR**1296134****[K1]**Bernhard Kawohl,*On a family of torsional creep problems*, J. Reine Angew. Math.**410**(1990), 1–22. MR**1068797**, 10.1515/crll.1990.410.1**[K2]**Bernhard Kawohl,*Rearrangements and convexity of level sets in PDE*, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR**810619****[LM]**A. C. Lazer and P. J. McKenna,*Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis*, SIAM Rev.**32**(1990), no. 4, 537–578. MR**1084570**, 10.1137/1032120**[L1]**Peter Lindqvist,*Stability for the solutions of 𝑑𝑖𝑣(\vert∇𝑢\vert^{𝑝-2}∇𝑢)=𝑓 with varying 𝑝*, J. Math. Anal. Appl.**127**(1987), no. 1, 93–102. MR**904212**, 10.1016/0022-247X(87)90142-9**[L2]**Peter Lindqvist,*On the equation 𝑑𝑖𝑣(\vert∇𝑢\vert^{𝑝-2}∇𝑢)+𝜆\vert𝑢\vert^{𝑝-2}𝑢=0*, Proc. Amer. Math. Soc.**109**(1990), no. 1, 157–164. MR**1007505**, 10.1090/S0002-9939-1990-1007505-7**[L3]**Peter Lindqvist,*Note on a nonlinear eigenvalue problem*, Rocky Mountain J. Math.**23**(1993), no. 1, 281–288. MR**1212743**, 10.1216/rmjm/1181072623**[L4]**P. Lindqvist,*On a nonlinear eigenvalue problem: stability and concavity*, Helsinki University of Technology, Inst. of Math. Research Reports # A279.**[R]**Paul H. Rabinowitz,*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**845785****[Sz]**Andrzej Szulkin,*Ljusternik-Schnirelmann theory on 𝐶¹-manifolds*, Ann. Inst. H. Poincaré Anal. Non Linéaire**5**(1988), no. 2, 119–139 (English, with French summary). MR**954468****[T]**P. Tolksdorf,*On the Dirichlet problem for quasilinear equations in domains with conical boundary points*, Comm. PDE 8 (1983), 773-817.

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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/S0002-9939-97-03961-0

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