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Journal of the London Mathematical Society 2005 72(3):645-662; doi:10.1112/S0024610705006848
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© The London Mathematical Society

A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

Wei Dong

Hebei University of Engineering Handan, Hebei 056021, China; School of Mathematics and Computer Science, University of New England Armidale, NSW 2351, Australia wdongau{at}yahoo.com.cn

Received 8 June 2004. Revision received 28 January 2005.

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem

Formula

where {Omega} sub RN (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*={infty} 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)+{varepsilon}, {varepsilon}>0. The results are strikingly different from those obtained for the case {alpha}=p–1.


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