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 -Delta(p)u = lambda u(a) - a(x)u(q) in Omega, u vertical bar partial derivative Omega = 0, where Omega subset of R-N (N >= 2) is a bounded smooth domain, p > 1 and lambda is a parameter, alpha,q are given constants such that p - 1 < alpha < p* - 1, alpha < q, p* = Np/(N - p) if N > p and p* = infinity 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) + epsilon, epsilon > 0. The results are strikingly different from those obtained for the case alpha = p - 1.