The Existence Of Positive Solutions For A Quasilinear Nonhomogeneous Elliptic Equation

In this paper,we study the existence of positive solution to the following quasilinear nonhomogeneous elliptic problem: where1<p<N,p*=NP/N-P is the critical Sobolev exponent,θ>0is a constant, D1,p(RN)is the closure of space C0∞(RN)under the norm andThe first main result of the paper reads:Theorem3.2Let P satisfy the conditions(P1)-(P4) and there exists a φ∈P, such that∫fφ>0.Then there exist θ>0,ρ0>0,such that if0<θ<θ0,then there exist u∈Mρ0,such that(1) and(2)(DIθ(u),v),v)≥0,(?)v∈P.By the conditions of theorem3.2,we see that P has many choices.In particular, let P be a collection of non-negative function,then we can prove that problem(0.1) has at least a positive solution.That is,Theorem3.3Let P={u∈X|u(x)≥0,a.e.x∈RN} and suppose f is a nonnegative function.Then there exist θ0>0,ρ0>0,such that if0<θ<θ0,then there exist u∈Mρ0,such that u is a weak positive solution of the problem(0.1)and...