The Study Of Multiple Solutions For Elliptic Problems

The appearance of perturbation on certain elliptic problems takes a rather remark-able importance to have global results where it equips several perspectives.The latter pushes us to deal with several problems with methods of perturbation in critical point theory where the main interest is based on the argument of compactness which is not valid in the general sense at least in a simple way.For this,the purpose of this dissertation is to discuss abstract methods with their applications to several perturba-tion problems whose common feature is to involve nonlinear problems on RN with a variational structure.For the following,our thesis contains four chapters presented as follows:·Chapter one:We summarize the background of the related problems and state the main results of the present thesis.Chapter two:We consider a class of Kirchhoff problems as follows-(a+b?RN |?u|2)?u=(1+?K(x))u2*-1,u>0 in RN,where a,b>0 are given constants,?is a small parameter,2*=2N/N-2,N>3 and K:R?R a function.First,we prove the non-degeneracy of positive solutions when ?=0.Then,as an application,vsing a perturbed argument and finite dimensional reduction method,we prove the existence of positive solutions for ?small.·Chapter three:Our interest is to prove the existence of positive solutions of the following nonlinear Kirchhoff equation with perturbed external source terms where a,b are positive constants,V(x),Q(x)are positive radial potentials,1<P<5,?>0 is a small parumeter,f(x)is an external source term in L2(R3)?L?(R3).Under the assumptions on V and Q and by applying the well-known Lyapunov-Schmidt reduction scheme and Lusternik-Schnirelman theory,we prove the existence of positive solutions to our problem.·Chapter four:We are concerned with the following system linearly coupled by nonlinear fractional elliptic equations where,for a(may be chosen to be s or t)in(0,1),(-?)? is the fractional Laplacian,??>-?1,?(?),?1,?(?)is the first eigenvalue of((-?)?,H0?(?)),2?*=2N/N-2?is a fractional Sobolev exponent,N>max{2s,2t} and(3 is a coupling parameter,?is a smooth bounded domain in RN.Using a variational method,we prove that this system has a positive ground state solution solution for,?>0.Via a perturbation argument,we also show that this system admits a positive higher energy solution when |?| is small.Moreover,the asymptotic behaviors of the positive ground state and higher solutions as ??0 are analyzed.