Fractional elliptic systems with critical nonlinearities

Author:

Bhakta Mousomi,Chakraborty Souptik,Miyagaki Olimpio H,Pucci Patrizia

Abstract

Abstract This paper deals with existence, uniqueness and multiplicity of positive solutions to the following nonlocal system of equations: }0\quad \text{in}\hspace{2pt}{\mathbb{R}}^{N},\end{aligned}\right.\qquad \qquad \qquad \qquad (\mathcal{S})\end{equation*}?> ( Δ ) s u = α 2 s * | u | α 2 u | v | β + f ( x ) in R N , ( Δ ) s v = β 2 s * | v | β 2 v | u | α + g ( x ) in R N , u , v > 0 in R N , ( S ) where 0 < s < 1, N > 2s, α, β > 1, α + β = 2N/(N − 2s), and f, g are nonnegative functionals in the dual space of H ˙ s ( R N ) , i.e., ( H ˙ s ) f , u H ˙ s 0 , whenever u is a nonnegative function in H ˙ s ( R N ) . When f = 0 = g, we show that the ground state solution of ( S ) is unique. On the other hand, when f and g are nontrivial nonnegative functionals with ker(f) = ker(g), then we establish the existence of at least two different positive solutions of ( S ) provided that f ( H ˙ s ) and g ( H ˙ s ) are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais–Smale sequences of the above system.

Funder

Università degli Studi di Perugia

National Board for Higher Mathematics

CNPq/Brazil

Istituto Nazionale di Alta Matematica \‘Francesco Severi\’

Department of Science and Technology India

FAPESP/Brazil

Publisher

IOP Publishing

Subject

Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics

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