Multiple Solutions For Nonhomogeneous P-Harmonic Equations

We are concerned with the existence of multiple solutions for p-harmonic equations with the boundary value problemwhere Ω is a bounded smooth domain in R^N;p > 1 is a constant;p < q < p*;p* = Np/(N-2p) is the critical Sobolev exponent for the embedding W₀^2,p(Ω) L^p(Ω);W₀^2,p(Ω) is a standard Sobolev space;denotes the N-dimensionalLaplacian;f(x) ∈ (W₀^2,p(Ω))* is some given function and (W₀^2,p(Ω))* denotes thedual space of W₀^2,p(Ω).In this paper, we obtain the results about the existence of multiple solutions for nonhomogeous p-harmonic equations by Ekeland's variational principle and the Mountain-Pass lemma under some assumptions on f(x).The organization of this paper is as follows.In Section 1, as the introduction, we list the research situation in this field and major results in this paper relative to the multiple solutions of nonhomogeous p-harmonic equations.In Section 2, firstly, we prove the existence of the first solution for problem (1) by Ekeland's variational principle under the assumption (1.6). Then through the approximation argument we give the proof of Theorem 1.1.In Section 3, we verify that I(u), the corresponding variational functional to (1), satisfies (PS) condition at first. Then we show the existence of the second solution for problem (1) under the assumption (1.6) and complete the proof of Theorem 1.2.In Section 4, we verify that the minimization problem (2.16) can be achieved by some function.