| Abstract: |
| In the first part of the talk, we discuss about recent results for equations in $\mathbb R^N$ driven by $(p,N)$ elliptic operators of Marcellini type and involving critical Trudinger-Moser nonlinearities with exponential growth. Our variational equations present some difficulties due to the double structure of the elliptic part combined with the double lack of compactness at critical level. In order to state a multiplicity result, we introduce a tricky step analysis based on the application of a completely new Br\`ezis and Lieb type lemma for exponential nonlinearities.
In the second part of the talk, we present a nontrivial generalization of the $(p,N)$ equations into a vectorial system, involving a coupled exponential nonlinearity. This situation forces more restrictive assumptions on the coupled exponential term than the related scalar counterpart. In particular, we introduce conditions which allow us to prove that both components of the vectorial solutions are nontrivial and different. That is, the system does not reduce into a single equation. |
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