The Nehari Manifold Method For Semilinear Elliptic Equations And Systems

In this thesis, we consider three classes of semilinear elliptic systems or equations with concave-convex nonlinearities. The first one is where Ω (?) RN is a bounded domain,1<q <2, and α,β>1satisfying α+β=2*=2N/N-2(N≥3). Moreover,f, g, h:(?)â†'R are sign-changing functions. The second one is a degenerate elliptic equation with cone critical Sobolev exponent where1<q <2,2*=2N/N-2(N≥3). The domain B is [0,1) x X, where X (?) RN-1is a compact set.â–³B:=(x1(?)Xl)2+(?)x2+…+(?)2%xN is an elliptic operator with totally characteristic degeneracy on the boundary x1=0. The third one is an elliptic equation with critical Sobolev exponent where1<q<2,2*=2N/N-2(N≥3), λ> O and f:(?),(?)â†'R is a continuous function. We apply the Nehari manifold method and the Ljusternik-Schnirelman theory to the first two classes of problems, and then obtain a relation between the number of positive solutions of them and the topology of the global maximum set of h or g. For the third problem, we use the Nehari manifold method and a mini-max principle to obtain that equation has at least four positive solutions when the domain is non-contractible.