Existence Of Global Solutions For A Class Of N - Laplacian Equations

This paper mainly studies the following nonlinear elliptic problems: where function F:Râ†'R is continuous and has critical growth, i.e.,f behaves like exp(α|u|N-1N), when|u|â†'∞. W1,N(RN) denotes the Sobolev space of functions in LN(RN) such that their weak derivatives are also in LN(RN) with the norm‖u‖NW1∫RN(|â–½u|N+|u|N)dx.The existence of the solution of the following problem have been considered by do O in [8]: where a(x) is coercive, i.e., a(x)â†'∞ as|x|â†'∞.The existence of nontrivial solution of the following equation have been studied by Lam and Lu in [9]: where V(x) is coercive or [V(x)]-1∈(RN).Coercive conditions play a very important role in [8] and [9]. Compared with the existing work, based on the research results, the different conditions is given, that is, in the absence of coercive condition, using Truinger-Moser inequality and mountain-pass geometry to prove the existence of nontrivial solutions of problem (1.1).Our main results are the following,Theorem1.1Under one of the following conditions,(f1) the function f:Râ†'R is continuous and for all u∈R, for some constants a0,b1, b2>0,where (f2) there is a constantμ> N such that, for all x∈RN and u>0,(f3) there are constants Ro, M0>0such that, for all x∈RN and u> R0,(f4) lim uf(u uniformly on compact subsets of RN(f5) lim sup uâ†'0uniformly for x∈RN.In the Sobolev space, there is a constant a0>0such that,(f6) There exist such that1, for every u>0. where Cq>0denote the best constant of Soboley embeddings.Then the problem (1.1) has a nontrivial weak solution u...