Existence Of Positive Solutions Of P(x)-Laplacian Equations In R～N

This paper studies the existence of positive solutions of a kind of p(x)-Laplacian with radially symmetric p(x) in R^N.That is,we consider the following nonlinear p(x)-Laplacian equation.-Δp(s)u=λ(x)g(u) in R^Nwhereλ∈C(R^N),λ≥0; g∈C((0,∞)),g≥0.And g is locally Holder continuous function.And p(x) is radially symmetry Holder continuous function,that is p(x) = p(|x|) = p(r) and 1 < p^-≤p(|x|)≤p⁺ <∞,where p⁺ = supp(x),p^-= inf p(x).we consider the the existence of solutions with property lim_{|x|→∞} u(x) ="positive constant".So it's not 0-boundary problem.Because of it we can not use the space W^1,p(x)(R^N) to deal with the problem.So we consider the space W_loc^1,p(x)(R^N).Because there have not norm in this space,we can not use variational methods.In this paper we take super-sub-solution methods to deal with the problem.Firstly,we consider the existence of solutions of the equation with A being a radially symmetric function.And we find some properties of the positive bounded solutions.Secondly,we consider the existence of solutions with generalλ.In this paper we take super-sub-solution to deal with this problem.We extending super-sub-solution argument to p(x) problems and construct the super- sub solutions via the solutions of the first case. Then we get our main conclusions.That is, this kind of equation has infinitely many bounded positive solutions,and these solutions are bounded from below.And each of the solutions tends to a positive constant at infinity.