Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

Abstract

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: {falsenonefalsearrayarrayleft−Δu−λ1u|x|2=a1|u|2*−2u+bhMathClass-open(xMathClass-close)αα+β|u|α−2u|v|β,arrayleftx∈RN,arrayleft−Δv−λ2v|x|2=a2|v|2*−2v+bhMathClass-open(xMathClass-close)βα+β|u|α|v|β−2v,arrayleftx∈RN, where N ≥ 3,λ1,λ2 ∈ [0,ΛN), ΛNMathClass-punc:MathClass-rel=()NMathClass-bin−222 is the best Hardy constant. 2MathClass-bin*MathClass-rel=2NNMathClass-bin−2 is the critical Sobolev exponent. a1,a2, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. h(x)MathClass-rel∈LqMathClass-rel′(double-struckRN) with qMathClass-rel′MathClass-rel=2MathClass-bin*2MathClass-bin*MathClass-bin−q. By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a1,a2,b and λ1,λ2. Copyright © 2012 John Wiley & Sons, Ltd.

The Brezis–Nirenberg problem for fractional systems with Hardy potentials

“…By variational arguments, Abdellaoui et al obtained the existence of different types of positive solutions to system (1.3). Since the work of Abdellaoui et al, 7 many results were obtained in this regard and we refer the reader to see, for instance, previous studies for the whole space [8][9][10] and in bounded domains. [11][12][13][14] Contrary to the local case that has been widely investigated, in recent years, several studies have been given to the fractional elliptic problems with singular Hardy potential.…”