On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in W01,p(Ω), by which we obtain the positive and negative cones of W01,p(Ω) are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.