The Brezis–Nirenberg problem for nonlocal systems

Abstract

Abstract. By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

On a systems involving fractional Kirchhoff‐type equations and Krasnoselskii's genus

“…The proof of Theorem 1.2 follows similar arguments used in the lemmas 2.3 and 3.1 contained in Faria et al, 20 combined with the theorem 1.2 and corollary 1.6 found in Ros-Oton and Serra. 35 Hence, to show the Theorem 1.2, it is enough to write the system (1) as follows:…”

“…We refer the reader to Colasuonno and Pucci, 16 Corrêa and Costa, 17 Corrêa and Figueiredo,18 and the references therein for Kirchhoff-type problems in others Sobolev spaces. For systems involving the fractional operator, without the presence of the Kirchhoff's term, results of existence and multiplicity can be found in Bai, 19 Faria et al, 20,21 Fiscella et al, 22 Fu et al, 23 Miyagaki and Pereira, 24 Wang et al, 25 and results involving concave-convex nonlinearities we quote W. Chen and Squassina, 26 He and Squassina, 27 and references therein.…”

Existence results for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

“…In the nonlocal case, Mukherjee and Sreenadh in [25] considered nonlocal counterpart of problem (1.4) and obtained existence, multiplicity and nonexistence results for solutions. Coming to the system of equations, elliptic systems involving fractional Laplacian and critical growth nonlinearities have been studied in [10,11,21,24], extending the Brézis and Nirenberg results for variational systems. Particularly, in [24], Miyagaki and Pereira studied the following fractional elliptic system…”

“…where λ, δ > 0 are real parameters and 1 < q < 2. Motivated by paper [11,24], we discuss the existence and multiplicity results for problem (1.1) under the conditions that (i) ξ 1 = ξ 2 = 0, 1 < p, q < 2 * µ , (ii) ξ 1 = ξ 2 = 0, p = q = 2 * µ , (iii) ξ 1 , ξ 2 > 0, p = q = 2 * µ respectively. The following are the main results.…”

Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity

“…Guo et al in [17] studied a nonlocal system involving fractional Sobolev critical exponent and fractional Laplacian. We also cite [5,10,29] as some very recent works on the study of fractional elliptic systems. However there is not much literature available on fractional elliptic system involving Choquard type nonlinearity.…”