On the eigenvalues of the $p$-Laplacian with varying $p$
HTML articles powered by AMS MathViewer
- by Yin Xi Huang PDF
- Proc. Amer. Math. Soc. 125 (1997), 3347-3354 Request permission
Abstract:
We study the nonlinear eigenvalue problem \begin{equation*}-\div (| \nabla u|^{p-2} \nabla u)=\lambda |u|^{p-2}u \quad \text {in}\; \Omega , \quad u=0\quad \text {on}\; \partial \Omega ,\tag *{(1) }\end{equation*} where $p\in (1,\infty )$, $\Omega$ is a bounded smooth domain in $\pmb R^{N}$. We prove that the first and the second variational eigenvalues of (1) are continuous functions of $p$. Moreover, we obtain the asymptotic behavior of the first eigenvalue as $p\to 1$ and $p\to \infty$.References
- J.P. G. Azorero and I.P. Alonso, Existence and uniqueness for the $p$-Laplacian: nonlinear eigenvalues, Comm. PDE 12 (1987), 1389–1430.
- Manuel del Pino, Manuel Elgueta, and Raúl Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u’|^{p-2}u’)’+f(t,u)=0,\;u(0)=u(T)=0,\;p>1$, J. Differential Equations 80 (1989), no. 1, 1–13. MR 1003248, DOI 10.1016/0022-0396(89)90093-4
- 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
- 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, DOI 10.1007/BF00256932
- 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, DOI 10.1007/978-3-642-61798-0
- 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, DOI 10.1090/S0002-9947-1988-0965762-2
- 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
- Bernhard Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. 410 (1990), 1–22. MR 1068797, DOI 10.1515/crll.1990.410.1
- Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
- 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, DOI 10.1137/1032120
- Peter Lindqvist, Stability for the solutions of $\textrm {div}\,(|\nabla u|^{p-2}\nabla u)=f$ with varying $p$, J. Math. Anal. Appl. 127 (1987), no. 1, 93–102. MR 904212, DOI 10.1016/0022-247X(87)90142-9
- Peter Lindqvist, On the equation $\textrm {div}\,(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164. MR 1007505, DOI 10.1090/S0002-9939-1990-1007505-7
- Peter Lindqvist, Note on a nonlinear eigenvalue problem, Rocky Mountain J. Math. 23 (1993), no. 1, 281–288. MR 1212743, DOI 10.1216/rmjm/1181072623
- P. Lindqvist, On a nonlinear eigenvalue problem: stability and concavity, Helsinki University of Technology, Inst. of Math. Research Reports # A279.
- 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, DOI 10.1090/cbms/065
- Andrzej Szulkin, Ljusternik-Schnirelmann theory on $\textit {C}^1$-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 119–139 (English, with French summary). MR 954468
- P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. PDE 8 (1983), 773-817.
Additional Information
- Yin Xi Huang
- Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
- Email: huangy@mathsci.msci.memphis.edu
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3347-3354
- MSC (1991): Primary 35P30, 35B30
- DOI: https://doi.org/10.1090/S0002-9939-97-03961-0
- MathSciNet review: 1403133