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Variational analysis for nonlocal Yamabe-type systems

  • * Corresponding author: Giovanni Molica Bisci

    * Corresponding author: Giovanni Molica Bisci 

Dedicated to Professor Patrizia Pucci with deep esteem and admiration

M. Xiang was supported by the National Natural Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program. G. Molica Bisci is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM-GNAMPA Projects Problemi variazionali su varietà Riemanniane e gruppi di Carnot (Prot 2016 000421) and Teoria e modelli per problemi non locali. B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199)

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  • The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider

    $\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$

    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.

    Mathematics Subject Classification: Primary: 35R11, 35A15; Secondary: 35J60.


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