Multiple non-trivial solutions of the Neumann problem for p-Laplacian systems
We obtain multiple non-trivial solutions of the Neumann problem for p-Laplacian systems using Morse theory.
Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri
Abstract. In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ & $ \mbox{in $\Omega$}\\ {\partial u \over \partial \nu} = 0 $ & $ \mbox{on $\partial \Omega$}\end{array} \right. $¶¶ where $ \nu $ is the outward unit normal to the boundary $ \partial\Omega $ of the bounded open set $ \Omega \subset \mathbb{R}^N $. The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.

On the Existence of a Non-trivial Solution for the $p$-Laplacian Equation with a Jumping Nonlinearity
. We consider the existence of a non-trivial weak solution for the equation (cid:1) − ∆ p u = f(x,u) in Ω , u = 0 on ∂Ω , where f satisﬁes f(x,u) = au p − 1 + − bu p − 1 − + o( | u | p − 1 ) ( p > 1) at 0 or ∞ . By using Morse theory and calcu-lating the critical groups, we show the existence of a non-trivial weak solution to the equation under mild auxiliary conditions.