Multiplicity of solutions for some semi-linear elliptic systems with non-homogeneous boundary conditions

In this article, we show that the existence of infinitely many weak solutions for some typical semi-linear elliptic systems governed by two Laplacian operators with non-homogeneous boundary conditions. A suitable perturbative method and variational tools will apply to such a non-symmetric problem: where Ω ⊂ R^N, (N ≥ 3) is a smooth bounded domain. The results of existence are obtained when and α,  satisfy some conditions.