The Existence Of Solutions To A Class Of Quasilinear Equations

In the first part of this article,we mainly study the existence of the solution for the quasilinear equation △pu+ φ(x,u)= 0,where △p represents the p-Laplacian operator,1<p<N,N ≥ 3.φ(x,u)is an appropriate continuous function on Rn ×(0,∞),which satisfies |φ(x,u)|≤ρ(x)f(u),where ρ is a positive continuous function on RN and f is a positive continuous function on(0,∞).We find that this problem is closely related with the existence of the solution for the below equation,This leads us to find a sufficient condition for the existence of bounded positive solu-tions for the equation △pu +φ(x,u)= 0.Since p-Laplacian operator is invariant about translation and rotaion,this sufficient condition is invariant under the isometry group G of RN.In the second part of this article,we study the existence and nonexistenc of large solutions for equation △pu = p(x)f(u),where ρ(x)is a nontrivial nonnegative con-tinuous function on RN,f is a nondecreasing positive function on(0,∞).Under the premise of the existence of bounded solution in RN for equation-△pu=ρ(x),we get a result about the nonexistence of large solution for equation △pu = ρ(x)f(u).The methods and theories we use in this paper are mainly the upper and lower-solutions methods and comparison principles.