In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schrödinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.2010 Mathematics Subject Classification. 35Q55; 35Q40.

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We study the convolution operators Tμ acting on the group algebras L1(G) and M (G), where G is a locally compact abelian group and μ is a complex Borel measure on G. We show that a cotauberian convolution operator Tμ acting on L1(G) is Fredholm of index zero, and that Tμ is tauberian if and only if so is the corresponding convolution operator acting on the algebra of measures M (G), and we give some applications of these results.