We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation urn:x-wiley:dummy:mana201200315:equation:mana201200315-math-0001 where , f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.
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Furtado et al. (2014) studied this question.