This paper is concerned with the study of the following double phase equation with logarithmic nonlinearity \(\begin{aligned}&-\operatorname {div}\left( |\nabla u|^{p-2}\nabla u+\mu (x) |\nabla u|^{q-2}\nabla u\right) + |u|^{p-2}u+\mu (x) |u|^{q-2}u \\&=K_{1}(x)|u|^{p^*-2}u+\lambda K_{2}(x)|u|^{r-2}u\log (|u|)\quad \text {in } \mathbb {R}^N, \end{aligned}\) with dimension \(N\ge 2\) , parameter \(\lambda >0\) , \(1<p<q<N\) , \(\mu :\mathbb {R}^{N}\rightarrow [0,\infty )\) is a Lipschitz continuous function and \(\max \{p,N(p-1)/(N-p)\}<r<p^*=Np/(N-p)\) . Here, the weight function \(K_1\) is positive, while \(K_2\) may change sign on \(\mathbb {R}^{N}\) . By a different variational approach, we prove an existence result which in some aspects improves our contribution in [A. Bahrouni, A. Fiscella, P. Winkert, J. Math. Anal. Appl. 547 (2025), no. 2, Paper No. 129311, 24 pp.]. For this, we need some restrictive assumptions on the weights \(\mu (\cdot )\) , \(K_1\) and \(K_2\) .
