The Existence and Non-Existence of Sign-Changing Solutions to Bi-Harmonic Equations with a p-Laplacian

Abstract

We investigate the bi-harmonic problem

$$\left\{ {\begin{array}{*{20}{c}} {{\Delta ^2}u - \alpha \nabla \cdot (f(\nabla )) - \beta {\Delta _p}u = g(x,u)}&{in}&\Omega \\ {\frac{{\partial u}}{{\partial u}} = 0,\frac{{\partial (\Delta u)}}{{\partial n}} = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{on}&{\partial \Omega } \end{array}} \right.$$

where ∆²u = ∆(∆u), ∆_pu = div (∣∇u∣^p−2∇u) with p > 2. Ω is a bounded smooth domain in ℝ^N, N ≥ 1. By using a special function space with the constraint ∫_Ωudx = 0, under suitable assumptions on f and g(x, u), we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.