Variational analysis for nonlocal Yamabe-type systems

Abstract

The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider \begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$\end{document} where $ (-\Delta )^s $ is the fractional Lapalcian, $ [u]_{s} $ is the Gagliardo seminorm of $ u $, $ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $ is a continuous function satisfying certain assumptions, $ V(x) = {|x|^{-2s}} $ is the Hardy potential function, $ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $, $ s, t\in (0, 1) $, $ \lambda, \mu $ are two positive parameters, $ 1<p<2_s^* = {6}/{(3-2s)} $ and $ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $. By using topological methods and the Krasnoleskii's genus theory, we obtain the existence, multiplicity and asymptotic behaviour of solutions for above problem under suitable positive parameters $ \lambda $ and $ \mu $. Moreover, we also consider the existence of nonnegative radial solutions and non-radial sign-changing solutions. The main novelties are that our results involve the possibly degenerate Kirchhoff function and the upper critical exponent in the sense of Hardy–Littlehood–Sobolev inequality. We emphasize that some of the results contained in the paper are also valid for nonlocal Schrödinger–Maxwell systems on Cartan–Hadamard manifolds.