The Research On The Existence Of Segregated Solutions For Nonlinear Elliptic Systems

In this thesis,we mainly use the gluing method and Lyapunov-Schmidt reduction method to study the existence of segregated solutions to several nonlinear elliptic systems under different parameter variations.The present thesis consists of five chapters:In chapter one,we summarize the background of the related problems and state main results of the present thesis.In chapter two,we construct a smooth radial positive solution for the following m-coupled elliptic system for β>0 large enough,where f∈ C2,1[0,+∞),f is an odd function in the sense of odd extension,B1(0)? is the unit ball centered at the origin,m≥ 3,N≥1 are positive integers.Our main result is an extension of Casteras and Sourdis[13]from m=2 to general case m>3 under some natural and essential non-degeneracy conditions by gluing method.The way we construct is somehow different and greatly simplify the computations since we overcome the difficulties brought by too much parameters from multiple equations.In chapter three,we consider the elliptic system-Δui=ui3+(?)βijuiuj2 in R4,i=1,...,q+1 when α:=βij and β:=βi(q+1)=β(q+1)j for any i,j=1,...,q.If β<0 and |β|is small enough we build solutions such that each component u1,…,uq blows-up at the vertices of q polygons placed in different great circles which are linked to each other,and the last component uq+1 looks like the radial positive solution of the single equation.In chapter four,we find infinitely many non-radial solutions for a system of Schrodinger equations with critical growth in a fully attractive or repulsive regime in presence of an external radial trapping potential,i.e.,This result uses the method of Local Pohozaev’s identity in[59]to construct concentrated solution,and overcomes the difficulties brought by the new coupling term.In chapter five,inspired by the above three chapters,we build a nonradial solution to the system-Δui+V(x)vi=ui3+β(?)uiuj2,x ∈R4,i=1,...,m as m is large enough.To the best of our knowledge,this is the first result about the existence of concentrated solutions to a system taking the number of components as a parameter.