Multiplicity results for a nonlocal system with critical growth

Jeziel N. Correia () and Claudionei P. Oliveira ()
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Jeziel N. Correia: Federal University of Pará
Claudionei P. Oliveira: Federal University of the South and Southeast of Pará

Partial Differential Equations and Applications, 2025, vol. 6, issue 4, 1-15

Abstract: Abstract In this paper, we will study the following fractional system with critical growth $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s} u + V(x)u = \displaystyle \frac{1}{2^{*}_{s}}Q_{u}(u,v)\quad \text {in } \mathbb {R}^{N},\\ \\ (-\Delta )^{s} v + W(x)v= \displaystyle \frac{1}{2^{*}_{s}}Q_{v}(u,v)\quad \text {in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ ( - Δ ) s u + V ( x ) u = 1 2 s ∗ Q u ( u , v ) in R N , ( - Δ ) s v + W ( x ) v = 1 2 s ∗ Q v ( u , v ) in R N , where $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$N \ge 4s$$ N ≥ 4 s , $$2^{*}_{s}=\displaystyle \frac{2N}{N-2s}$$ 2 s ∗ = 2 N N - 2 s and V and W are potential functions. We combine a recent global compactness result by Correia and Oliveira [1] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.

Keywords: Fractional elliptic system; Variational methods; Krasnoselskii genus; Critical exponent; Primary: 35A15; 35J50; Secondary: 35R11; 58E05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00337-y

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