In this paper, we establish the existence of nontrivial solutions for a (p, q)-Laplacian equation characterized by: \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u = \lambda |u|^{p-2}u + u_{+}^{p^{*}-1} + g(x,u_{+}) + f(x) & \text {in } \Omega \\ u=0 & \text {on } \partial \Omega , \end{array} \right. \end{aligned}\) where \(\Omega \) denotes a bounded domain with a smooth boundary within \({\mathbb {R}}^N\) , the parameters satisfy \(1<q<p<N\) , and g represents a nonlinearity exhibiting subcritical growth while f belongs to \(L^{\infty }(\Omega )\) . This equation, inspired by the seminal Ambrosetti-Prodi problems. Our approach leverages the application of variational methods, alongside precise estimates for Talenti functions pertinent to the critical issue at hand. We demonstrate the existence of two distinct solutions under the condition that the nonlinearity crosses the initial eigenvalue (i.e., \(\lambda <\lambda _{1}\) , while also a unilateral critical growth in \(+\infty \) is considered).
