Contain Caffarelli Kohn - Nirenberg Critical Index Of Studies Of Singular Nonlinear Elliptic Equations

In this thesis, by using variational methods, the critical point theory and theperturbation arguments, we study the critical singular problems of the Ca?arelli- Kohn-Nirenberg type, and we obtain a series of new existence and multiplicity results. Ourresults are contained in the chapters 2-6 of this thesis.In Chater 2, we mainly consider the existence of G-symmetric solutions for thesingular nonlinear elliptic equations. Specifically, (1) we deal with existence and mul-tiplicity of positive G-symmetric solution of semilinear singular elliptic equations inRN . The main di?culty here lies in the critical Ca?arelli-Kohn-Nirenberg exponentand the unboundedness of the domain. The results obtianed here extend the corre-sponding ones in [1, 2]; (2) we deal with the existence and multiplicity of G-symmetricsolutions for some quasilinear elliptic equations with critical Hardy-Sobolev exponentsin a bounded domain ? ? RN. The di?culty here mainly lies in proving suitablesymmetry critical point principle and dealing with the lower order perturbation terms.The results obtianed here extend the corresponding ones in [1, 2, 3].In Chapter 3, we are concerned with the existence of positive solutions for somesingular nonlinear elliptic equations with small parameters, and our results include:(1) when the exponent of the perturbation term uγsatisfies 0 <γ< 1, we obtainthe existence of two positive solutions for the problem (Qγ) by using the minimaxmethods and analytic techniques; (2) when the exponentγof the perturbation termuγequals 1, we prove the existence of one positive solution for the problem (Q?1) byusing perturbation methods. The results here extend the corresponding ones in [4, 5].In Chapter 4, we study the existence of positive solutions for the singular quasilin-ear elliptic equations with nonhomogeneous terms. Our results obtianed in this chapterextend the corresponding ones in [6, 7].In Chapter 5, we discuss the Brezis-Nirenberg problem with the critical Ca?arelli-Kohn-Nirenberg exponents. In particular, (1) we deal with existence and multiplicityof positive solutions for some semilinear singular elliptic equations with indefinite po-tentials in bounded domain ?. Our results unify and extend the the correspondingones in [8]; (2) we deal with some Brezis-Nirenberg type problem of the quasilinearsingular elliptic equation. By using the Lusternik-Schnirelmann category theory, weobtain the multiplicity result for the problem. Our results extend the corresponding ones in [9, 10, 11].In Chapter 6, as the last part of this thesis, we mainly consider existence of radialsolutions for a class of singular quasilinear elliptic equations with critical Ca?arelli-Kohn-Nirenberg exponents. The di?culty lies in the loss of the compactness for thecorresponding functional. Via variational method and analytic technique, we overcomethis di?culty and obtain the existence and multiplicity results of the radial positiveand nodal solutions for the equations. These results extend the corresponding ones in[12, 13, 14, 15].