Extremals for fractional order Hardy–Sobolev–Maz’ya inequality

Abstract

In this article, we derive the existence of positive solution of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.Note that, existence of nontrivial solution of (1.1) will follow for the case of s = 1, if we can show the existence of minimizers of (1.4). For β = 0, the existence of minimizers of (1.4) has been established in [2] by using concentration compactness principle due to P.L Lions [25], [24]. Whereas, for 0 < β < (k−2) 2 4 , the existence is proved in [29], by using blow up analysis for approximate solutions with a rescaling argument. On the other hand, since for β = (k−2) 2 4 , the expected space in which the minimizers will belong is much bigger than the same for the case of β < (k−2) 2 4 , one needs to employ a careful analysis. Using a penalty method, Tintarev and Tertikas proved the existence of minimizers for the case of β = (k−2) 2 4 in [33] and subsequently improved in [19].The cylindrical symmetry of the local counterpart (i.e. s = 1) of (1.1), has been established in [26] by using moving plane method in the special case of β = 0. In fact, when t = 1, they have classified all the solutions by a careful asymptotic analysis. Subsequently, in the case of 0 ≤ β < (k−2) 2 4

Weak Comparison Principle for Weighted Fractional p -Laplacian Equation

“…For some abstract definitions and tools of fractional Laplace operator, see [13]. For more recent results of fractional Laplace elliptic problem, see [14][15][16] and the reference therein.…”

On fractional Orlicz–Sobolev spaces

“…The last two decades have witnessed an increasing number of investigations on these spaces because of their use in the analysis of nonlocal elliptic and parabolic equations, whose study has received an enormous impulse in the same period -see e.g. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][24][25][26][27]31,34,35,[38][39][40][43][44][45][46][47][48][49][50][51][52][53][54]56,57,59,60,64]. The aim of this note is to survey a few recent results, contained in [1][2][3][4], on some aspects of fractional Orlicz-Sobolev spaces.…”

Fractional Orlicz-Sobolev embeddings

“…This has been favoured by a myriad of investigations on nonlocal equations of elliptic and parabolic type, whose solutions naturally belong to the spaces W s,p (R n ). A touch of recent contributions in this connection is furnished by [7,8,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26,27,28,29,41,44,45,51,52,53,58,59,61,62,63,64,65,68,70,71,77,76,78,83,86]. Comprehensive treatments of the theory of fractional Sobolev spaces, as special instances of the more general Besov spaces, can be found e.g.…”