Abstract. In this paper we establish the reversed sharp Hardy-LittlewoodSobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in [8]) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.

We study the existence of solutions of the following nonlinear Schrödinger equationWe assume that 0 does not lie in the spectrum of −∆+V and µ < (N −2) 2 4 , N ≥ 3. The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small µ ≥ 0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If µ < 0 and 0 lies below the spectrum of −∆ + V , then ground state solutions do not exist.MSC 2010: Primary: 35Q55; Secondary: 35J10, 35J20, 58E05

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The paper is concerned with the slightly subcritical elliptic problem with Hardy-critical termin a bounded domain

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