Combined effects in coupled quasilinear elliptic systems via a “like” Gagliardo–Nirenberg inequality

In this paper, we study the existence, nonexistence and multiplicity results to a class of quasilinear elliptic system involving ‐Laplacian by exploiting the competition of nonlinearities. We mainly propose a “like” Gagliardo–Nirenberg inequality for a pair of functions, and use it to combine the variational method to prove the existence, nonexistence and multiplicity of nontrivial nonnegative solutions for a quasilinear elliptic system involving ‐Laplacian under different conditions between parameters.