Thomas Bartsch, Zhaoli Liu and Tobias Weth

We prove that the $p$-Laplacian problem

$-\Delta_p u = f(x, u)$, with $u \in W^{1, p}_0 (\Omega)$

on a bounded domain $\Omega \subset R^N$, with $p > 1$ arbitrary, has a nodal solution provided that $f : \Omega \times R \to R$ 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.

Keywords: $p$-Laplacian equation, superlinear non-linearity, nodal solution, variational method, invariant set of descending flow.

E-mail:
Thomas.Bartsch@math.uni-giessen.de
zliu@mail.cnu.edu.cn
Tobias.Weth@math.uni-giessen.de