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Proceedings of the London Mathematical Society 2005 91(1):129-152; doi:10.1112/S0024611504015187
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© London Mathematical SOciety

Nodal Solutions of a p-Laplacian Equation

Thomas Bartsch, Zhaoli Liu and Tobias Weth

Mathematisches Institut, Universität Giessen Arndtstrasse 2, 35392 Giessen, Germany. E-mail: Thomas.Bartsch{at}math.uni-giessen.de, Tobias.Weth{at}math.uni-giessen.de
Department of Mathematics, Capital Normal University Beijing 100037, P. R. China. E-mail: zliu{at}mail.cnu.edu.cn

Received 18 November 2003.

We prove that the p-Laplacian problem –{Delta}p u = f(x, u), with u isin Formula on a bounded domain {Omega} sub RN, with p > 1 arbitrary, has a nodal solution provided that f : {Omega} x R -> R is subcritical, and f(x, t) / |t|p – 2 is superlinear. Infinitely many nodal solutions are obtained if, in addition, f(x, –t) = –f(x, t). 2000 Mathematics Subject Classification 35J20, 35J65, 58E05.

Key Words: p-Laplacian equation • superlinear non-linearity • nodal solution • variational method • invariant set of descending flow

The research of the second author was supported by the AvH in Germany and NSFC: 10441003.


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