Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator

Abstract

Abstract By using an abstract critical point theorem based on a pseudo-index related to the cohomological index, we prove the bifurcation results for the critical Choquard problems involving fractional p-Laplacian operator: (−Δ)psu=λ|u|p−2u+(∫Ω|u|pμ,s∗|x−y|μdy)|u|pμ,s∗−2uin Ω,u=0in RN∖Ω, $$(- \Delta)_{p}^{s} u = \lambda \vert u \vert ^{p-2} u + \biggl( \int_{\Omega}\frac{ \vert u \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\,dy \biggr) \vert u \vert ^{p_{\mu,s}^{*}-2}u \quad \text{in } \Omega,\qquad u = 0\quad \text{in } {\mathbb {R}}^{N} \setminus \Omega, $$ where Ω is a bounded domain in RN ${\mathbb {R}}^{N}$ with Lipschitz boundary, λ is a real parameter, p∈(1,∞) $p\in(1,\infty)$, s∈(0,1) $s\in (0,1)$, N>sp $N>sp$, and pμ,s∗=(N−μ2)pN−sp $p_{\mu,s}^{*}=\frac{(N-\frac{\mu}{2})p}{N-sp}$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. These extend results in the literature for the fractional Choquard problems, and they are still new for a p-Laplacian case.

Keywords

• Fractional p-Laplacian
• Choquard equation
• Critical exponent
• Z 2 ${\mathbb {Z}}_{2}$ -cohomological index
• Pseudo-index