| Abstract: |
| In this talk, we discuss about double phase equations set in $\mathbb R^N$ involving critical Sobolev terms and logarithmic perturbations. By variational methods, we provide different existence results for our equations. The main difficulties arise from the presence of the logarithmic perturbation, which is sign-changing, combined with a double lack of compactness, due to the free action of translation group in $\mathbb R^N$ and the critical Sobolev nonlinearity. Furthermore, we have to deal with the Luxemburg type norm of the solution space, which complicates even the study of geometry for the energy functional. Our results, contained in https://doi.org/10.1016/j.jmaa.2025.129311 and in https://doi.org/10.1007/s00030-025-01172-1, are new even in the classical $p$-Laplacian case. |
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