Multiple Solutions Of Some Nonlinear Elliptic Equations With Fast Increasing Weight And Critical Exponent

In this paper,we study three classes of nonlinear elliptic equations with fast increasing weight and critical growth.Our main results are summarized as follows.1.We study a class of elliptic equation with fast increasing weight and concave-convex nonlinearities:-div（K（x）u）=a（x）K（x）|u|^q-2u+b（x）K（x）|u|^2*-2u,x∈R^N,（1P）where N≥3,1<q<2,K（x）=exp（|x|^α/4）withα≥2.With some slightly stronger conditions than the ones presented in[42],we obtain the existence of two positive solutions and a pair of sign-changing solutions by applying Nehari manifold method and Ekeland variational principle.Our result supplements and comple-ments the main result in[42]considering the existence of nontrivial solutions for the problem（1P）.2.We study the following Kirchhoff type problem with fast increasing weight and concave-convex nonlinearities:-a+R3K（x）|u|2dxdiv（K（x）u）=λK（x）f（x）|u|q-2u+K（x）|u|4u,x（2∈P）R3,where 1<q<2,K（x）=exp（|x|^α/4）withα≥2,>0 is small enough,and the parameters a,λ>0.Under some assumptions on f（x）,we establish the existence of two positive nontrivial solutions and obtain uniform lower estimates for extremal values of the problem（2P）.Our result improves the main result in[59]concerning the existence of two positive solutions for the Kirchhoff type problem with concave-convex nonlinearities and critical growth.3.We study a class of elliptic equation with fast increasing weight and critical growth:-div K（x）u=λK（x）|x|^β|u|^q-2u+Q（x）K（x）|u|^2*-2u,x∈R^N,（3P）where N≥3,2<q<2^*,λ>0 is a parameter,K（x）=exp（|x|^α/4）,α≥2,β=（α-2）_（2*-2）^（2*-q）and 0≤Q（x）∈C（R^N）.We prove that the problem（3P）has at least k positive solutions and k sign-changing solutions,provided that the maximum of Q（x）is achieved at k different points.Our result can be regarded as an extension of[17]considering the effect of the shape of Q（x）on the existence of nontrivial solutions,in the whole space.