ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35, 34 Bognár, Gabriella, University of Miskolc, Hungary On a nonlinear eigenvalue problem The nonlinear diffusion can be described by the nonlinear partial differential equation ∑_{i=1}^{N} \frac{∂}{∂x_{i}} (|∇u|^{p−1}∇u) + λ|u|^{p−1}u = 0 in D ∈ R^{N}. We can consider it as a generalization of the linear partial differential equation Δu + λu = 0. Another generalization of the linear equation is the following equation

∑_{i=1}^{N} \frac{∂}{∂x_{i}} [|\frac{∂u}{∂x_{i}} |^{p−1} \frac{∂u}{∂x_{i}} ] + λ|u|^{p−1}u = 0

in D ∈ R^{N}, where 0 < p < ∞. Our purpose is to show that the nonlinear eigenvalue problem in bounded domain has eigensolutions and corresponding eigenvalues. The classical solution for different domains D will be given when D is bounded by a rectangle or by a central symmetric convex curve. References: F. de Thelin, Quelques résultats d’existence et de non-existence pour une E. D. P. elliptique non linéarie, C. R. Acad. Sci. Paris, 299, Série I. Math., pp. 911-914 (1984). W. Reichel, W. Walter, Radial solutions of equations and inequalities involving the p-Laplacian, J. of Inequal. and Appl., 1, pp. 47–71 (1997).