On eigenvalue problems of $p$-Laplacian with Neumann boundary conditions
HTML articles powered by AMS MathViewer
- by Yin Xi Huang PDF
- Proc. Amer. Math. Soc. 109 (1990), 177-184 Request permission
Abstract:
We study the nonlinear eigenvalue problem \[ - {\Delta _p}u = \lambda m(x)|u{|^{p - 2}}u \quad {\text {in}} \Omega ,\quad \frac {{\partial u}} {{\partial n}} = 0 \quad {\text {on}} \partial \Omega , \;{\text {where}} p > 1,\lambda \in {\mathbf {R}}.\] For $\int _\Omega {m(x) < 0}$, we prove that the first positive eigenvalue ${\lambda _1}$ exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case $\int _\Omega {m(x) = 0}$, we prove that ${\lambda _0} = 0$ is the only eigenvalue with a positive eigenfunction.References
- Aomar Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725–728 (French, with English summary). MR 920052
- C. Bandle, M. A. Pozio, and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z. 199 (1988), no. 2, 257–278. MR 958651, DOI 10.1007/BF01159655
- E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5
- 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
- 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
- Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690, DOI 10.1080/03605308008820162
- P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions, Nonlinear analysis and optimization (Bologna, 1982) Lecture Notes in Math., vol. 1107, Springer, Berlin, 1984, pp. 106–114. MR 778584, DOI 10.1007/BFb0101496 J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
- Mitsuharu Ôtani and Toshiaki Teshima, On the first eigenvalue of some quasilinear elliptic equations, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 1, 8–10. MR 953752
- Stefan Senn and Peter Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann. 258 (1981/82), no. 4, 459–470. MR 650950, DOI 10.1007/BF01453979
- Peter Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 (1983), no. 7, 773–817. MR 700735, DOI 10.1080/03605308308820285
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 177-184
- MSC: Primary 35P15; Secondary 35J65
- DOI: https://doi.org/10.1090/S0002-9939-1990-1010800-9
- MathSciNet review: 1010800