Abstract :
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.
