On A Class Of Elliptic Equation And System Involving Critical Exponent

The thesis is considered with the existence of multiple solutions for semilinear elliptic equation and quasilinear equation or system involving critical Sobolev exponent.In chapter two, by concentration-compactness principle, we consider the existence of mul-tiple positive solutions for the following problem whereλ>0 is a parameter,2*=2N/N-2 is the critical Sobolev exponent and N>4, f(x,t) and h(x)are some given functions. We prove that there existsλ*∈(0,+∞) such that problem (a) has exactly two positive solutions forλ∈(0,λ*) by Barrier method and Mountain Pass Lemma and no positive solutions forλ>λ*. Moreover, ifλ=λ*, problem(a)has a unique solution (λ*,uλ*),which means that (λ*,uλ*) is a turning point in H1(RN)for problem(a).In chapter three, we consider multiple positive solutions for the following p-Laplace elliptic equation involving concave-convex nonlinearities and a Hardy-type term whereΩ(?)RN is a bounded domain such that 0∈Ω,1<q<p,λ>0,μ<μ=(N-p/p)pis the best Hardy constant,p*=Np/N-p is the critical Sobolev exponent.f and g are nonnegative functions. By extracting the Palais-Smale sequence in the Nehari manifold,the existence of multiple positive solutions to this equation whenλdifferent range is verified.In the last chapter, by an approximation argument and Pohozaev identity, we obtain the existence of infinitely many high positive energy radial solutions for the following elliptic system with critical Sobolev growth: where B(?)RN is a unit ball with the center at origin. N>2(p+q+p+1)/p+q-1,σ>0,p,q,α,β>1 andα+β=2* where 2*=2N/N-2,2≤p+q<2*.