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Fast decaying ground states for elliptic equations with exponential nonlinearity. (English) Zbl 1455.35101

For superlinear continuous functions \(f\), with exponential subcritical or exponential critical growth, the authors use a Trudinger-Moser inequality and lemmas involving the Nehari manifold, to propose conditions under which \[-\operatorname{div}(\exp(|x|^2/4) \nabla u) = \exp(|x|^2/4) f(u) \quad \mbox{in}\quad \mathbb{R}^2\] has a positive minimal energy solution.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
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