A PRIORI ESTIMATES AND EXISTENCE OF POSITIVE SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem −Δp u = λuα − a(x)uq in Ω, u|∂Ω = 0, where Ω ⊂ RN (N 2) is a bounded smooth domain, p > 1 and λ is a parameter, α,q are given constants such that p − 1 < α < p∗ − 1, α < q, p∗ = Np/(N − p) if N > p and p∗ = ∞ when N p, and a(x) is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where a(x) is replaced by a(x)+ , > 0. The results are strikingly different from those obtained for the case α = p − 1.