Sign-changing solutions for a critical exponential problem with competing potentials - Barahona, Willy; Lapa, Eugenio Cabanillas; Figueiredo, Giovany M. (Topological Methods in Nonlinear Analysis, 2025)
We establish the existence, concentration, and exponential decay of a family of sign-changing solutions for a problem involving exponential critical growth, described by the equation: $$ \begin{cases} -\epsilon^{2}\mbox{div}\left(a(x) abla u\right)+V(x)u =K(x) f(u) & \mbox{in $\mathbb{R}^2,$}\\ u \in H^{1}\big(\mathbb{R}^{2}\big). \end{cases} \leqno{(\rom{P}_{\epsilon})} $$ To address the lack of compactness and the competition between the potentials, we employ variational methods alongside a suitable truncation of the nonlinearity.