Multiple nontrivial solutions for a double phase system with concave-convex nonlinearities in subcritical and critical cases

In this article, we study the double phase elliptic system which contain with the parametric concave-convex nonlinearities and critical growth. The introduction of mixed critical terms brings some difficulties to the problem. For example, in proving that the solution is nontrivial, we need to do an additional series of studies on scalar equation. By introducing a new optimal constant \(S_{\alpha ,\beta }\) $S_{\alpha ,\beta }$ in the double phase system, considering the different magnitude relationships of the exponential terms, and using the fibering method in form of the Nehari manifold and the Brezis-Lieb Lemma, the existence and multiplicity of solutions in subcritical and critical cases are obtained separately.