Qualitative Properties Of Solutions For Several Semilinear Elliptic Problems In The Whole Space

In this dissertation,we mainly study the existence,non-existence and other qual-itative properties of solutions of several kinds of semilinear elliptic equations,includ-ing symmetry,conformal invariance and uniform boundedness,etc.The full text is divided into four chapters.In Chapter 1,we investigate the existence and nonexistence of positive solutions for the semilinear elliptic systems where n? 3,p,q>0 and max{p,q}?1.We obtain the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case,and the sufficient and necessary conditions for the classification of solutions in critical case,and the Joseph-Lundgren-type condition for existence of local stable solutions.In Chapter 2,we study the existence and non-existence of positive solutions for a weighted Laplace equation-?u+?/|x|2u=uq,u>0 in Rn\{0},where ??-(n-2)2/4 and q>1.We prove the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case,and a classification result in the critical case and the invariant properties of the quasi-commonality.Finally,we establish the Joseph-Lundgren-type condition for existence of stable solutions.In Chapter 3,we consider the priori estimate of an equation of the Chern-Simons-Higgs type.We study the uniform bound of classical solutions u of the semilinear equation and integrable solutions u of the fractional order equation.We prove |u| ?1 in Rn,which plays an important role in studying the quantization effects of those equations.In Chapter 4,we study the optimal functions of several kinds of non-local in-equalities,including the Hardy-Littlewood-Sobolev inequality,fractional Gagliardo-Nirenberg inequality,nonlocal Gagliardo-Nirenberg inequality and Coulomb-Sobolev inequality.Firstly,we derive the Euler-Lagrange equations which they satisfy.Sec-ondly,we investigate the existence of some integrable classical solutions for these equations,where the Pohozaev identity plays a key role.One of the characteristics of the coupled equations studied in Chapter 1 is that the right-hand term is homogeneous with respect to u and v,so we think that the qualitative properties of the solutions are close to that of the well-known Lane-Emden equation.In Chapter 2,the equations with Hardy-Leray potential are quasi-conformal invariant,which should be close to the related properties of Lane-Emden equation.So we verify that the two kinds of problems are identical in the critical exponents which are used to study the existence.When dealing with the classification of solutions in the critical case in Chapter 1,we introduce the Newton potential and transform differential equations into integral equations.In fact,this idea can be used to deal with equations containing nonlocal Laplace operators.As an application,in Chapter 3,we investigate a priori estimate of solutions of a fractional Chern-Simons-Higgs equation.As a comparison,we also give a priori estimate of the Chern-Simons-Higgs equation with a second-order Laplace operator by using the comparison principle.In Chapter 4,we consider the non-local equations with Riesz potential.Obvious-ly,the Riesz potential is a fractional extension of the Newton potential.They are all related to the best functions of nonlocal Gagliardo-Nirenberg inequalities.We derive their Euler-Lagrange equations with non-local terms and give the necessary condi-tions for the existence of solutions,which correspond to the Liouville theorem in the subcritical cases in Chapter 1 and Chapter 2.