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

In this paper, we study the 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. In this paper, we obtain the results by Fibering Method and Variational Principle under some assmuptions on h(x).The main conclusions in this paper are listed as followFirst conclusion, let 2≤p <+∞if N = 2,3 or (?) < p <+∞if N≥4; q > p. Then for any h(x)∈L¹(Ω) with ||h||₁ small enough system (*) has at least three solutions (?), such thatSecond conclusion, let 2≤p≤(?), and 0＜(?)<(?) if N≥4. Then for any h(x)∈L^γ(Ω) with ||h||_γsmall enough system (*) has at le(?)st three solutions (?), such thatWhereγ=(?).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 multiple solutions for nonhomogeneous 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. These lemmas play a important role in proving the chief theorem in this paper.In Section 3, we prove that the equation (*) has at least three solutions under some different assumptions on the perturbation h(x).