Existence And Concentration Of Positive Solutions For Quasilinear Choquard Equation With Doubly Critical Growth

As an important class of nonlinear partial differential equations,quasilinear Schrodinger equation plays an important role in quantum mechanics,fluid science and other fields.Choquard equation describes the electromagnetic wave propagation in plasma,and plays an important role in the condensation of Bose-Einstein theory and the practical physical problems.Therefore,it is of profound physical significance to study these problems.For the existence,multiplicity and concentration of quasilinear Choquard equations caused the wide attention of mathematicians.As a result of the existence of quasilinear term,the corresponding energy functional of this kind of equations in usually Sobolev space cannot be defined.Therefore,by using variable substitution method,the critical point theory,the concentration-compactness principle and the mountain road theorem,we study the existence,multiplicity and concentration of positive solutions for quasilinear Choquard type equation.The main contents of this paper are as follows:Chapter 1 mainly introduces the research background,significance and research ststus of the quasilinear Schrodinger equations and the quasilinear Choquard type equations.In Chapter 2,we give some basic function spaces,related definitions,lemmas and theorens.In Chapter 3,we study the existence,multiplicity and concentration of positive solu-tions for the following quasilinear Choquard equations-?2?u+V(x)u-?2?(u2)u=?2(I2*|u|10)|u|8u+|u|10u+h(u),x?R3,where ?>0,I2=1/4?|x|is a Riesz potential.V?C(R3,R),h?C1(R,R)satisfy the following conditions:(V)0<V0:=infx?R3V(x)<lim|x|??V(x):=V?<?;(h1)h(s)=0,s?0;lims??h(s)/sq-1=0,q?(6,12);(h2)|h'(s)|?C(1+|s|q-2);(h3)there exists ??(10,12)such that 0<?H(s)=??0sh(t)dt?sh(s),s>0;(h4)s?h(s)/s5 is an increasing function in(0,+?).We make the changing of variables,that is,we consider v=f-1(u),where f is defined by#12 f(t)?-f(-t),t?(-?,0].As above quasilinear equation is transformed into a semilinear elliptic equation,the cor-responding energy functional can be defined in Sobolev space.Then,we consider the ex-istence of positive solutions of the limit equation,calculate the non-compactness level of the corresponding energy functional,and estimated that the energy level is below the non-compactness level.Finally,the existence,multiplicity and concentration of the positive solution of the equation are obtained by using the variational method,the critical point theory,the concentration-compactness principle and the Ljusternik-Schnirelmann theory.