The Infinite Multiple Solutions For The P-harmonic Equations With The Navier Boundary Value Conditions

In this paper, we study the infinite multiple solutions for the following p-harmonic equations with the Navier boundary value conditions :whereΩis a smooth bounded domain in R^N,â–³denotes N-dimensional Laplacian. W₀^1,p(Ω) and W^2,p(Ω) are the standing Sobolev Space.Let 2≤p < +∞if N = 2,3 or (?)< p <+∞if N≥4. Then, system (*) has infinite multiple solutions. Let 2≤p≤(?),0 <(?)<(?), if N≥4. Then, system (*) also has infinite multiple solutions.In this paper, we obtain the results about the existence of infinite multiple solutions for p-harmonic equations by Fibering Method and Variational Principle under some assumptions on f(u).The organization of this paper is as follow.In Section 1, as the introduction, we list the research situation in this field and major results in this paper relative to infinite multiple solutions for p-harmonic equations.In Section 2, we prove some basic lemmas about the bifurcation equation and variational functional which are associated with (*) by Fibering Method and Variational Principle.In Section 3, we prove that the equation (*) has infinite multiple solutions under some different assumptions about the perturbation f(u).