On Multiplicity of Positive Solutions for Quasilinear Equation with Co-normal Boundary Condition

Abstract

Abstract Let Ω ⊂ ℝN, N ≥ 3, be a bounded domain with C2 boundary, , the critical exponent for the Sobolev imbedding. In this work, we are interested in the following problem: where λ > 0, 0 ≤ q < p - 1. We show that there exists 0 < Λ < ∞ such that for suitable ranges of p and q, (Pλ) admits at least two solutions in W1,p (Ω) if λ ∈ (0, Λ) and no solution if λ > Λ. The proof of these assertions is done by first finding the local minimum for the variational functional associated to (Pλ) and then applying mountain pass arguments to obtain a saddle point type solution. In the critical case we are considering, there are technical reasons which make the mountain pass argument work for only certain ranges of p and q.