Nehari Manifold for Fractional Kirchhoff Systems with Critical Nonlinearity

Abstract

In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff system $$\begin{aligned} {\left\{ \begin{array}{ll}\mathcal {L}_{M}(u) = {\lambda }f(x)|u|^{q-2}{u} + \frac{2{\alpha }}{{\alpha }+{\beta }} |u|^{\alpha -2} \,u|v|^{\beta} &{}\quad \mathrm{in}\, \Omega ,\\ \mathcal {L}_M(v) = {\mu }g(x)|v|^{q-2}v + \frac{2{\beta }}{{\alpha }+{\beta }}|u|^{\alpha} \,|v|^{\beta -2}v &{}\quad \mathrm{in}\, \Omega ,\\ \quad \;\;\; u = v = 0 &{}\quad \mathrm{on}\, {\partial }{\Omega }, \end{array}\right. } \end{aligned}$$ where $$\mathcal{L}_{M}(u) = M \big(\int_{\Omega} |(-{\Delta})^\frac{s}{2}u|^{2}dx\big) (-\Delta)^{s}u$$ is a double non-local operator due to a Kirchhoff term $$M(t) = a + bt$$ with a, b > 0 and the fractional Laplacian $$(-\Delta)^{s}, s \in (0,1)$$. We consider that $$\Omega$$ is an open and bounded domain in $$\mathbb{R}^{N}, 2s 0$$ are real parameters, $$1 < q < 2, \alpha,\beta \geq 2$$ and $$\alpha + \beta = 2_{s}^{*} = 2N/(N-2s)$$ is a fractional critical exponent. Using the idea of Nehari manifold technique and a compactness result based on the classical idea of the Brezis–Lieb lemma, we prove the existence of at least two positive solutions for $$(\lambda,\mu)$$ lying in a suitable subset of $$\mathbb{R}^{2}_{+}$$.