Abstract
In this paper, we study the following class of Schrödinger equations: \[ -\Delta_{N}u+V(|x|)|u|^{N-2}u=Q(|x|)h(u) \quad \text{in } \mathbb{R}^N, \] where $N\geq 2$, $V,Q\colon \mathbb{R}^{N}\rightarrow \mathbb{R}$ are potentials that can be unbounded, decaying or vanishing at infinity and the nonlinearity $h\colon \mathbb{R}\rightarrow \mathbb{R}$ has a critical exponential growth concerning the Trudinger-Moser inequality. By using a variational approach, a version of the Trudinger-Moser inequality and a symmetric criticality type result, we obtain the existence of nonnegative weak and ground state solutions for this class of problems and under suitable assumptions, we obtain a nonexistence result.
Citation
Yane Lísley Araújo. Gilson Carvalho. Rodrigo Clemente. "Quasilinear Schrödinger equations with singular and vanishing potentials involving nonlinearities with critical exponential growth." Topol. Methods Nonlinear Anal. 57 (1) 317 - 342, 2021. https://doi.org/10.12775/TMNA.2020.024
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