Existence And Nonexistence Of Nontrivial Solutions For A Class Of Critical Growth Polyharmonic Elliptic Systems

In this paper, we consider the existence and nonexistence of nontrivial solutions for the following polyharmonic elliptic systems involving critical Sobolev exponents:whereΩis a bounded smooth domain in R^N, m is a positive integer, 2^*=2N/(N-2m) isthe critical Sobolev exponents for the embedding is stan-dard Sobolev space, denotes the N-dimensional Laplacian, (—Δ)^mdenotes polyharmonic operator, A=(a_ij)∈R^n×n is a real symmetric matrix.According to the different scopes of m, p, we obtain the existence and nonexistence of nontrivial solutions for a class of polyharmonic elliptic systems by using Mountain-pass Lemma without (PS) condition and constrained variation.The organization of this paper is as follows:In section 1, we introduce the background associated with the multiple harmonic problems and the main results of this paper.In section 2, we discuss problem (1) with m = 2,p = 2. In this case, problem (1) can be reduced toWe prove the existence and nonexistence of the solutions for problem (2) with constrained variational method.In section 3, we show the existence and nonexistence of the solutions for problem (1) by using Mountain-pass Lemma without (PS) condition when m is a positive integer and 2 < p < 2^*.