Normalized Solutions For The Coupled Nonlinear Fractional Kirchhoff Type Systems

Kirchhoff equation is an important class of nonlocal partial differential equations with a wide range of applications in elasticity theory,biological systems,and nonNewtonian mechanics and so on.In this thesis,by using variational methods and critical point theories,we study the normalized solutions of the strongly coupled fractional Kirchhoff type systems.The main contents are as follows:Firstly,we review the latest research results for the coupled nonlinear systems and introduce preliminary results.Secondly,we are concerned with the existence of positive solutions for the following strongly coupled fractional Kirchhoff type systems in Hs(RN)× Hs(RN)(N=2,3):satisfying the normalized conditions ‖u‖L2(RN)2=C12 and ‖u‖L2(RN)2=c22,where ci>0 is prescribed,ai,bi,μi＞0,Vi(x):RN→R(i=1,2)is nonnegative potential,β is a coupling parameter,r1,r2>1 and 2<p1,p2,r1+ r2<2s*,N=2,3,s∈(N/4,1),2s*=2N/(N-2s)is the fractional Sobolev critical exponent.Under constant vanishing potentials and attractive interspecies interactions,two cases are studied:one is L2-subcritical and the other is L2-supercritical.In the first case,we prove the existence of a positive solution by the constrained minimizing methods.In the second case,by using a minimax procedure,we prove the existence of a mountain pass type ground state solution under perturbations of the coupling parameter.Moreover,we study the L2-critical case under certain type of trapping potentials.In this case,for β>0 and β<0,we prove the existence of a positive solution by introducing some auxiliary minimization problems.Finally,we pose some interesting questions for further exploration.0 picture,0 table,76 references...