In this article, we consider the following Gradient type (p, q)-Laplacian System \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} -\Delta _{p}u +|u|^{p-2}u& = \alpha |u|^{\alpha -2}|v|^{\beta }u,\\ -\Delta _{q} v +|v|^{q-2}v& = \beta |u|^{\alpha }|v|^{\beta -2}v \qquad \text{ in } \mathbb {R}^N, \end{aligned} \end{array}\right. } \end{aligned}\) with \(1<p,q<N\) , \(N\ge 4\) , and \(\alpha , \beta >1\) satisfying some suitable subcritical conditions. We prove the existence of sign-changing solutions to this system. Symmetry plays a crucial role for this result, and we prove the existence of solutions which are invariant under some specific group action. In the first part, using the Mountain-Pass Theorem and the Principle of Symmetric Criticality, a special case for dimensions \(N\ne 5\) is discussed. And, for the general result, we study a suitable Palais–Smale sequence to get the existence.
