Nonhomogeneous systems involving critical or subcritical nonlinearities

Fractional elliptic systems with critical nonlinearities

“…Furthermore, if f ≡ g, then the solution (u, v) of (S) has the property that uv, whenever α = β. Finally, if α = β but f ≡ g, then u ≡ v. Theorem 1.2 complements the mentioned work [2] on (S). The proof of the uniqueness theorem 1.1 is inspired by some arguments made in [15,35] (also see [16]).…”

“…It is well-known that u ∈ Ḣs (R N ) implies u ∈ L p loc (R N ) for any p ∈ [2, 2 * s ]. In the vectorial case, as described in [2], the natural solution space for (S) is the Hilbert space Ḣs (R N ) × Ḣs (R N ), equipped with the inner product (u, v), (φ, ψ) Ḣs × Ḣs := u, φ Ḣs + v, ψ Ḣs , and the norm (u, v) Ḣs × Ḣs := u 2 Ḣs + v 2 Ḣs 1 2 .…”

“…We refer to [3,14,19,24,28,30], where Dirichlet systems of equations in bounded domains have been treated. For the nonlocal systems of equations in the entire space R N , we cite [2,25,26] and the references therein. In the very recent work [2], the first, second and fourth authors of this current paper have proved existence of one solution to (S) when f and g are nontrivial but f ( Ḣs ) and g ( Ḣs ) are small enough.…”

“…For the nonlocal systems of equations in the entire space R N , we cite [2,25,26] and the references therein. In the very recent work [2], the first, second and fourth authors of this current paper have proved existence of one solution to (S) when f and g are nontrivial but f ( Ḣs ) and g ( Ḣs ) are small enough. To the best of our knowledge, so far there have been no papers in the literature, where uniqueness/multiplicity of positive solutions have been established for (S), with the fractional Laplacian and the critical exponents in R N .…”

“…where N > 2s, α, β > 1 and α + β < 2 * s , and f , g are nonnegative functionals in the dual space of H s (R N ) (see [2] for existence of solutions). It is known that the scalar equation…”