New weighted sharp Trudinger–Moser inequalities defined on the whole euclidean space \( {\mathbb {R}}^N \) and applications


In this paper, we provide an extension to the whole euclidean space \( {\mathbb {R}}^N,\ N \ge 2, \) of the Trudinger–Moser inequalities proved by Calanchi and Ruf (Nonlinear Anal 121:403–411, 2015) involving a logarithmic weight. The inequalities are new and highlight very well the importance of the presence of this type of weight. Next, we prove some version of the concentration-compactness principle due to P.L. Lions giving some new improvements of the Trudinger–Moser inequalities established in the first part of this work. In the light of this last result, we treat some elliptic quasilinear problems involving new type of exponential growth condition at infinity.

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Correspondence to Sami Aouaoui.

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Aouaoui, S., Jlel, R. New weighted sharp Trudinger–Moser inequalities defined on the whole euclidean space \( {\mathbb {R}}^N \) and applications. Calc. Var. 60, 50 (2021).

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Mathematics Subject Classification

  • 26D15
  • 35A15
  • 35B33
  • 35D30
  • 35J20
  • 35J62
  • 46E35