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On the eigenvalues of the \(p\)-Laplacian with varying \(p\). (English) Zbl 0882.35087

Summary: We study the nonlinear eigenvalue problem \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= \lambda|u|^{p-2}u\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(p\in(1,\infty)\), \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\). 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 \(p\to1\) and \(p\to\infty\).

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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[1] J.P. G. Azorero and I.P. Alonso, Existence and uniqueness for the \(p\)-Laplacian: nonlinear eigenvalues, Comm. PDE 12 (1987), 1389-1430. · Zbl 0637.35069
[2] Manuel del Pino, Manuel Elgueta, and Raúl Manásevich, A homotopic deformation along \? of a Leray-Schauder degree result and existence for (|\?’|^{\?-2}\?’)’+\?(\?,\?)=0,\?(0)=\?(\?)=0,\?>1, J. Differential Equations 80 (1989), no. 1, 1 – 13. · Zbl 0708.34019
[3] 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. · Zbl 0595.35100
[4] Henrik Egnell, Existence and nonexistence results for \?-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal. 104 (1988), no. 1, 57 – 77. · Zbl 0675.35036
[5] 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. · Zbl 0562.35001
[6] Mohammed Guedda and Laurent Véron, Bifurcation phenomena associated to the \?-Laplace operator, Trans. Amer. Math. Soc. 310 (1988), no. 1, 419 – 431. · Zbl 0713.34049
[7] 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. · Zbl 0818.34013
[8] Bernhard Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. 410 (1990), 1 – 22. · Zbl 0701.35015
[9] Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. · Zbl 0593.35002
[10] 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. · Zbl 0725.73057
[11] Peter Lindqvist, Stability for the solutions of \?\?\?(|∇\?|^{\?-2}∇\?)=\? with varying \?, J. Math. Anal. Appl. 127 (1987), no. 1, 93 – 102. · Zbl 0642.35028
[12] Peter Lindqvist, On the equation \?\?\?(|∇\?|^{\?-2}∇\?)+\?|\?|^{\?-2}\?=0, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157 – 164. · Zbl 0714.35029
[13] Peter Lindqvist, Note on a nonlinear eigenvalue problem, Rocky Mountain J. Math. 23 (1993), no. 1, 281 – 288. · Zbl 0785.34050
[14] P. Lindqvist, On a nonlinear eigenvalue problem: stability and concavity, Helsinki University of Technology, Inst. of Math. Research Reports # A279. · Zbl 0838.35094
[15] 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. · Zbl 0609.58002
[16] Andrzej Szulkin, Ljusternik-Schnirelmann theory on \?\textonesuperior -manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 119 – 139 (English, with French summary). · Zbl 0661.58009
[17] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. PDE 8 (1983), 773-817. · Zbl 0515.35024
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