Constrained And Free Variational Problems To Several Elliptic Equations And Systems

In this thesis,we mainly consider the existence and asymptotic properties of normalized solutions to the Sobolev critical Kirchhoff equations and Gross-Pitaevskii equations,the existence of nontrivial weak solutions to a doubly critical equation involving a fractional Laplacian with a Hardy term,and also the improved Sobolev inequalities in product Sobolev spaces and its applications to doubly critical elliptic systems involving fractional Laplacian and Hardy terms.The thesis consists of five chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we consider the existence and asymptotic properties of nor-malized solutions to the following Kirchhoff equation-(a+b?RN |?u|2)?u=?u+|u|p-2u+?|u|q-2u,x?R3 where a>0,b>0,2<q<14/3<p?6 or1/3<q<p?6,?>0 and ??R appears as a Lagrange multiplier.In both cases for the range of p and q,the Sobolev critical exponent p=6 is involved and the corresponding energy functional is unbounded from below on the L2-spheres.If 2<q<10/3 and 14/3<p<6,we obtain a multiplicity result to the equation.If 2<q<10/3<p=6 or 14/3<q<p?6,we get a normalized ground state to the equation.Furthermore,we derive several asymptotic results on the obtained normalized solutions.Our results extend the results of N.Soave(J.Differential Equations 2020&J.Funct.Anal.2020),which studied the nonlinear Schrodinger equations,to the Kirchhoff equations.In Chapter Three,we consider the existence,asymptotic behavior and stability of normalized ground states to the Gross-Pitaevskii equation-1/2?u+?1|u|2u+?2(K*|u|2)u+?3|u|p-2u+?u=0,x? R3,where 2<P<10/3,(?1,?2,?3 ? R2×R-,*denotes the convolution,K(x)=1-3cos2?(x)/|x|3?(x)is the angle between(0,0,1)and x? R3,and ?? R appears as a Lagrange multiplier.When the physical parameters describing the strength of nonlinearities lie in a defined range,the corresponding energy functional is unbounded on the L2-spheres,so we turn to study a suitable local minimization problem and prove the existence of normalized ground states.Furthermore,we show that any normalized ground state is a local minimizer and it is stable under the Cauchy flow,which indicates that lower power(mass-subcritical)three-body losses stabilize the system that is initially unstable.Finally,by refining the upper bound of the normalized ground state energy,we provide a precise description of the asymptotic behavior of the normalized ground states as the mass vanishes.In Chapter Four,we consider the existence of nontrivial weak solutions to a doubly critical equation involving a fractional Laplacian with a Hardy term.Before proving the main result,we develop some useful tools based on a weighted Morrey space.The main results in this chapter have been published in(Acta Math.Sci.Ser.B(Engl.Ed.),40,1808-1830,2020).In Chapter Five,we study the improved Sobolev inequalities involving weighted Morrey norms in product Sobolev spaces and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms.The main results in this chapter have been published in(Discrete Contin.Dyn.Syst.Ser.S,doi:10.3934/dcdss.2020469).